
Class T/4 54 



Book 






Sb 



y Z 



A TREATISE 



ON 



SUEVETING 



COMPRISING THE THEORY AND 
THE PRACTICE 



BY 



WILLIAM M. GILLESPIE, LL. D. 

FORMERLY PROFESSOR OF CIVIL ENGINEERING IN UNION COLLEGE 



REVISED AND ENLARGED BY 

OADY STALEY, Ph. D. 

PEE8IDENT OF CASE SCHOOL OF APPLIED SCIENCE 




NEW YORK 
D. APPLETON AND COMPANY 

1887 






■■■' 



*S7 



Coptright, 1S55, 18S7, 
By D. APPLETON AND COMPANY. 



S~ ^<? 



PEEFAOE 



Gillespie's "Land-Surveying" was first printed in 1851, for 
use in Professor Gillespie's classes in Union College. It was 
published in 1855, and very soon became the standard authority 
on land-surveying. 

In the preface to the first edition Professor Gillespie says : 

"Land-surveying is perhaps the oldest of the mathematical 
arts. Indeed, geometry itself, as its name — 'land-measuring' 
— implies, is said to have arisen from the efforts of the Egyp- 
tian sages to recover and to fix the landmarks annually swept 
away by the inundations of the Nile. The art is also one of 
the most important at the present day, as determining the title 
to land, the foundation of the whole wealth of the world. It 
is, besides, one of the most useful as a study, from its striking 
exemplifications of the practical bearings of abstract mathemat- 
ics. But, strangely enough, surveying has never yet been re- 
duced to a systematic and symmetric whole. To effect this, by 
basing the art on a few simple principles and tracing them 
out into their complicated ramifications and varied applications 
(which extend from the measurement of 'a mowing -lot' to 
that of the heavens), has been the earnest endeavor of the 
present writer. 

"The work, in its inception, grew out of the author's own 
needs. Teaching surveying, as preliminary to a course of civil 
engineering, he found none of the books in use (though very 
excellent in many respects) suited to his purpose. He was, 
therefore, compelled to teach the subject by a combination of 



iv PREFACE. 

familiar lectures on its principles and exemplifications of its 
practice. His notes continually swelling in bull?:, gradually be- 
came systematized in nearly their present form. His system 
has thus been fully tested, and the present volume is the 
result. 

" A double object has been kept in view in its preparation : 
viz., to produce a very plain introduction to the subject, easy 
to be mastered by the young scholar or the practical man of 
little previous acquirement, the only prerequisites being arith- 
metic and a little geometry; and at the same time to make 
the instruction of such a character as to lav a foundation broad 
enough and deep enough for the most complete superstructure 
which the professional student may subsequently wish to raise 
upon it." 

In the preface to the " Land-Surveying," Professor Gilles- 
pie announced that another volume, on ''Leveling and Higher 
Surveying," was to follow. This work was, at the time of his 
death, in 1868, unfinished. 

The same method was pursued in its preparation as for the 
" Land-Surveying." Parts of it had been printed for class use, 
and a large part of the book had been given in the form of 
lectures to the Professor's classes. It was completed by the 
editor of this volume, and published in 1870. 

The two volumes, "Land-Surveying" and "Leveling and 
Higher Surveying," are now revised and united in this volume. 

The best authorities have been consulted, in order to render 
the work as reliable as possible. 

The writer is under obligations to mauy friends for assist- 
ance in the work of revision, and especially to E. P. Dickey. 
C. E., for a large part of " Mining- Surveying," and to Professor 
T. W. Wright, C. E., for the formula and table in gradienter 
measurement, and other valuable assistance. 

Cadt Staley. 

Case School of Applied Science, 
Cleteland, Ohio, January, 1SS7. 



GENERAL DIVISION OF THE SUBJECT. 

[A full Analytical Table of Contents is given at the end of the volume.] 



PART I. 



LAND-SURVEYING. 
CHAP. PAGE 

I. General Principles and Operations ........ 1 

II. Chain-Surveying . . . ... . . . , . 50 

lit. Compass-Surveying . 100 

IV. Transit-Surveying . 185 

V. Obstacles to Surveying 242 

VI. Laying out and dividing up Land 263 

VII. Surveying United States Public Lands ..... 301 

PART II. 

LEVELING. 

I. Direct Leveling 389 

II. Indirect Leveling . . , . . . , . . . . 385 
III. Barometric Leveling -. 399 

PART III. 

TOPOGRAPHY. 

I. First System — Contour-Lines ......... 408 

II. Second System — Hatchings . . . , , . . . .417 

III. Third System— Shades from Vertical Light 419 

IV. Conventional Signs 423 

The Plane-Table . . . .-■"; . . , . . .431 

PART IV. 

TRIANGULAR SURVEYING, 

I. Plane Surfaces ........... 442 

II. Geodesy ... 464 

PART V. 

HYDROGRAPHIOAL SURVEYING. 

I. The Sextant . . . ... . . . . , ,472 

II. Trilinear Surveying ...... » » 485 



vi GENERAL DIVISION OF THE SUBJECT. 

CHAP. PAGE 

III. Surveying the Shore-Line 489 

IV. Soundings ' 491 

V. The Chart 496 

PART VI. 

MININGr-SUEYETIXG-. 

I. Surveying Old Lines ' . . . . 498 

II. Locating New Lines 518 

Appendix A. — Synopsis of Plane Trigonometry 523 

Appendix B. — Transversals, Harmonic Division, etc. ..... 532 

Analytical Table of Contents 538 

Tables : 

Traverse-Tables. 

Table of Chords. 

Logarithms of Numbers. 

Logarithmic Sines, Cosines, Tangents, etc. 

Natural Sines, Cosines, Tangents, etc. 

Stadia-Table. 

Table of Refraction in Declination. 



PAET I. 
LAND-SURVEYING 



CHAPTER I. 

GENERAL PRINCIPLES AND FUNDAMENTAL OPERATIONS. 

DEFINITIONS AND METHODS. 

1. Surveying is the art of making such measurements as will 
determine the relative positions of any points on the surface of the 
earth ; so that a Map of any portion of that surface may be drawn, 
and its Content calculated. 

2. The position of a point is said to be determined, when it is 
known how far that point is from one or more given points, and 
in what direction therefrom ; or how far it is in front of them or 
behind them, and how far to their right or to their left, etc. ; so 
that the place of the first point, if lost, could be again found by 
repeating these measurements in the contrary direction. 

The " points " which are to be determined in Surveying are not 
the mathematical points treated of in Geometry, but the corners 
of fences, boundary stones, trees, and the like, which are mere 
points in comparison with the extensive surfaces and areas which 
they are the means of determining. In strictness, their centers 
should be regarded as the points alluded to. 

A straight Line is "determined," that is, has its length and 
its position known and fixed, when the points at its extremities 
are determined ; and a plane Surface has its form and dimensions 
determined when the lines which bound it are determined. Con- 
sequently, the determination of the relative positions of points is 
all that is necessary for the principal objects of Surveying ; which 



2 LAND-SURVEYWG. 

are to make a map of any surface, such as a field, farm, State, etc., 
and to calculate its content in square feet, acres, or square miles. 
The former is an application of Drafting, the latter of Mensuration. 
The position of a point may be determined by a variety of 
methods. Those most frequently employed in Surveying are the 
following — all the points being supposed to be in the same plane : 

3. First Method. By measuring the distances from the required 
point to tzvo given points. 

Thus, in Fig. 1, the point S is " determined," if it is known to 

Fig. l. be one inch from A, and half an inch from 

jfa t B ; for its place, if lost, could be found by 

,.'-'' \ describing two arcs of circles, from A and B 

,.--'' \ as centers, and with the given distances as 

A-^ ^B 

radii. The required point would be at the 

intersection of these arcs. 

In applying this principle in surveying, S may represent any 
station, such as a corner of a field, an angle of a fence, a tree, a 
house, etc. If, then, one corner of a field be 100 feet from a second 
corner, and 50 feet from a third, the place of the first corner is 
known and determined with reference to the other two. 

There will be two points fulfilling this condition, one on each 
side of the given line, but it will always be known which of them 
is the one desired. 

In Geography, this principle is employed to indicate the po- 
sition of a town ; as when we say that Buffalo is distant (in a 
straight line) 295 miles from New York, and 390 from Cincinnati, 
and thus convey to a stranger acquainted with only the last two 
places a correct idea of the position of the first. 

In Analytical Geometry, the lines A S and B S are known as 
" Focal Co-ordinates" the general name " co-ordinates " being ap- 
plied to the lines or angles which determine the position of a point. 

4. Second Method. By measuring the perpendicular distance 
from the required point to a given line, and the distance the nee 
along the line to a given point. 

Thus, in Fig. 2, if the perpendicular distance S C be half an 



DEFINITIONS AND METHODS. 3 

inch, and C A be one inch, the point S is " determined " ; for its 
place could be again found by measuring one inch from A to C, 
and half an inch from C, at right angles to FlG , 2 . 

AC, which would fix the point S. 

The public lands of the United States 
are laid out by this method, as will be ex- 



plained in . Chapter VII. A c 

In Geography, this principle is employed under the name of 
Latitude and Longitude. 

Thus, Philadelphia is one degree and fifty-two minutes of lon- 
gitude east of Washington, and one degree and three minutes of 
latitude north of it. 

In Analytical Geometry, the lines A C and C S are known as 
"Rectangular Co-ordinates." The point is there regarded as de- 
termined by the intersection of two lines, drawn parallel to two 
fixed lines, or "Axes, 1 '' and at a given distance from them. These 
Axes, in the present figure, would be the line A 0, and another 
line, perpendicular to it and passing through A, as the origin. 

5. Third Method. By measuring the angle between a given 

line and a line drawn from any given point 

S of it to the required point ; and also the 

s'' length of this latter line. 

,--'' Thus, in Fig. 3, if we know the angle 

A-^-1 B B AS to be a third of a right angle, and 

AS to be one inch, the point S is deter- 
mined ; for its place could be found by drawing from A, a line 
making the given angle with A B, and measuring on it the given 
distance. 

In applying this principle in surveying, S, as before, may rep- 
resent any station, and the line A B may be a fence, or any other 
real or imaginary line. 

In " Compass Surveying," it is a north-and-south line, the 
direction of which is given by the magnetic needle of the compass. 
In Geography, this principle is employed to determine the rela- 
tive positions of places, by " bearings and distances " ; as when 
we say that San Francisco is 1,750 miles nearly due west from St. 



4 LAND-SURVEYING. 

Louis ; the word " west'' indicating the direction, or angle which 
the line joining the two places makes with a north-and-south line, 
and the nnmber of miles giving the length of that line. 

In Analytical Geometry, the line A S, and the angle BAS, are 
called "Polar Co-ordinates." 

6. Fourth. Method. By measuring the angles made with a 
given line by two other lines starting from given points upon it, 

and passing through the required point. 
S Thus, in Fig. 4, the point S is deter- 

,.---'' \ mined by being in the intersection of the 

,,---''' \ two lines AS and B S, which make respect- 

j^ ^ ively angles of a half and of a third of a 

right angle with the line A B, which is one 
inch long ; for the place of the point could be found, if lost, by 
drawing from A and B lines making with A B the known angles. 
In Geography , we might thus fix the position of St. Louis, by 
saying it lay nearly due north from Xew Orleans, and clue west 
from "Washington. 

In Analytical Geometry, these two angles would be called 
"Angular Co-ordinates." 

7. In Fig. 5 are shown together all the measurements neces- 
sary for determining the same point S, by each of the four pre- 
ceding methods. In the First Method, 

we measure the distances A S and B S : in 

s 
the Second Method, the distances AC and ^'\ 

C S, the latter at right angles to the y- 

former ; in the Tliird Method, the dis- A .^l — -b 

tance A S, and the angle SAB; and, in 

the Fourth Method, the angles S A B and S B A. In all these 

methods the point is really determined by the intersection of two 

lines, either straight lines or arcs of circles. Thus, in the First 

Method, it is determined by the intersection of two circles ; in 

the Second, by the intersection of two straight lines ; in the 

Third, by the intersection of a straight line and a circle : and, in 

the Fourth, by the intersection of two straight lines. 



Fig. 5. 




DEFINITIONS AND METHODS. 5 

8. Fifth Method. By measuring the angles made with each 
other by three lines of sight passing from the required point to 
three points whose positions are known. 

Thus, in Fig. 6, the point S is determined by the angles A S B 
and B S C, made by the three lines S A, 
S B, and S 0. 

Geographically, the position of Chi- 
cago would be determined by three straight 
lines passing from it to Washington, Cin- 
cinnati, and Mobile, and making known 
angles with each other ; that of the first ~~~XTf~ 

and second lines being about one third, s© 

and that of the second and third lines, about one half of a right 
angle. 

From the three lines employed, this may be named the Method 
of Trilinear Co-ordinates. 

9. The position of a point is sometimes determined by the 
intersection of two lines, which are themselves determined by their 

extremities being giyen. Thus, in Fig. 7, 
the point S is determined by its being sit- 
uated in the intersection of A B and C D. 
This method is sometimes employed to fix 
the position of a station on a railroad line, 
etc., when it occurs in a place where a 
stake can not be driven, such as in a pond, 
and in a few other cases, but is not used frequently enough to 
require that it should be called a sixth principle of Surveying. 

10. These five methods of determining the positions of points 
produce five corresponding systems of Surveying, which may be 
named as follows : 

I. Diagonal Sueveying. 
II. Peependiculae Sueveyikg 

III. POLAE SlIEVEYING. 

IV. Teiangulae Sueveyi^g. 
V. Teilineae Sueyeyikg. 



6 LAXD-SURVEYING. 

The above division of Surveying has been made in harmony 
with the principles involved and the methods employed. 

The subject is, however, sometimes divided with reference to 
the instruments employed ; as the chain, either alone or with 
cross-staff ; the compass ; the transit or theodolite -; the sextant ; 
the plane-table, etc. 

11. Surveying may also be divided according to its objects. 
In Land Surveying, the content, in acres, etc., of the tract 

surveyed, is usually the principal object of the survey. A map, 
showing the shape of the property, may also be required. Certain 
signs on it may indicate the different kinds of culture, etc. This 
land may also be required to be divided up in certain proportions ; 
and the lines of division may also be required to be set out on the 
ground. One or all of these objects may be demanded in Land 
Surveying. 

In Topographical Surveying, the measurement and graphical 
representation of the inequalities of the ground, or its " relief," 
i. e., its hills and hollows, as determined by the art of "Level- 
ling," is the leading object. 

In Maritime or Hydrograpliical Surveying, the positions of 
rocks, shoals, and channels are the chief subjects of examination. 

In Mining Surveying, the directions and dimensions of the 
subterranean passages of mines are to be determined. 

12. Surveying may also be divided according to the extent of 
the district surveyed into Plane and Geodesic. Geodesy takes into 
account the curvature of the earth, and employs Spherical Trigo- 
nometry. Plane Surveying disregards this curvature, as a need- 
less refinement except in very extensive surveys, such as those of a 
State, and considers the surface of the earth as plane, which may 
safely be done in surveys of moderate extent. 

13. In all the methods of Land Surveying, there are three 
stages of operation : 

1. Measuring certain lines and angles, and recording them ; 

2. Drawing them on paper to some suitable scale ; 

3. Calculating the content of the surface surveyed. 



MAKING THE MEASUREMENTS. 



MAKING THE MEASUREMENTS. 

14. The Measurements which are required in Surveying may 
be of lines or of angles, or of both, according to the Method em- 
ployed. Each will be successively considered. 

Measuring Straight Lines. 

15. The lines, or distances, which are to be measured, may be 
either actual or visual. 

Actual lines are such as really exist on the surface of the land 
to be surveyed, either bounding it, or crossing it ; such as fences, 
ditches, roads, streams, etc. 

Visual lines are imaginary lines of sight, either temporarily 
measured on the ground, such as those joining opposite corners of 
a field ; or simply indicated by stakes at their extremities or other- 
wise. If long, they are " ranged out " by methods to be given. 

Lines are usually Fig. 

measured with chains, 
tapes, or rods, divided 
into yards, feet, links, 
or some other unit of 
measurement. 

16. Gunter's Chain. 

This is the measure 
most commonly used 
in Land Surveying. 
It is 66 feet, or 4 rods 
long.* Eighty such 
chains make one mile. 
It is composed of 
one hundred pieces of 
iron or steel wire, or 
links, each bent at the end into a ring, and connected with the 

* This length was chosen (by Mr. Edward Gunter) because 10 square chains of 66 
feet make one acre, and the computation of areas is thus greatly facilitated. For 
other surveying purposes, particularly for railroad work, a chain of 100 feet is pref- 
erable. On the United States Coast and Geodetic Survey the unit of measurement is 
the French Metre, equal to 3'281 feet nearly. 




8 



LAND-SUR V EYING. 



ring at the end of the next piece by another ring. Sometimes two 
or three rings are placed between the links. The chain is then less 
liable to twist and get entangled or "kinked." Two or more 
swivels are also inserted in the chain, so that it may turn around 
without twisting. Every tenth link is marked by a piece of brass, 
having one, two, three, or four points, corresponding to the number 
of tens which it marks, counting from the nearest end of the chain.* 
The middle or fiftieth link is marked by a round piece of brass. 

The hundredth part of a chain is called a link.f The great 
advantage of this is that, since links are decimal parts of a chain, 
they may be so written down, 5 chains and 43 links being 5 "4:3 
chains, and all the calculations resjiecting chains and links can 
then be performed by the common rules of decimal arithmetic* 
Each link is 7'92 inches long, being = 66 X 12 -f- 100. 

The following table will be found convenient : 



CHAINS INTO FEET. 




FEET INTO LINKS. 


Chains. 


Feet. 


Chains. 


Feet. 


Feet. 


Links. 


Feet. 


LiLts. 


001 


0-66 


1-00 


65- 


o-io 


0-15 


io- 


15-2 


0-02 


1-32 


2- 


132- 




0-20 


0-30 


15- 


22-7 


003 


1-98 


3- 


198- 




0-25 


0-38 


20- 


30-3 


0-04 


2-64 


4- 


264- 




0-30 


0-45 


n- 


37*9 


0-05 


3-30 


5- 


330- 




0-40 


0-60 


30- 


45-4 


0-06 


3-96 


6' 


396* 




0-50 


0'76 


33- 


50-0 


0-07 


4-62 


1- 


462- 




0-60 


0-91 


35- 


53-0 


0-08 


5-28 


s- 


528' 




070 


1-06 


40- 


606 


0-09 


5-94 


9- 


594- 




0-75 


113 


45- 


68-2 


o-io 


6 60 


10- 


660- 




0-80 
0*90 


1*21 

136 


50- 
55' 


75-S 
83-3 


0-20 


13-20 


20- 


1320- 




1-00 


1*62 


60- 


90-9 


0-30 


19-80 


30' 


1980- 




2- 


30 


65' 


98-5 


0-40 


26-40 


40- 


2640- 




3' 


4-5 


70- 


106-1 


0-50 


33-00 


50' 


3300- 




4' 


6-1 


75- 


113-6 


0-60 


39-60 


60- 


3960' 




6- 


7-6 


80- 


121-2 


0*70 


46 20 


70- 


4620' 




6- 


91 


85- 


1288 


0-80 


52-80 


80- 


5280- 




7- 


10-6 


90- 


136-4 


0-90 


59-40 


90- 


5940- 




8* 


12-1 


95- 


143-9 


1-00 


66-00 


100- 


6600- 




9- 


13-6 


100- 


151-5 



* To prevent the very common mistake of calling forty, sixty ; or thirty, seventy ; 
it has been suggested to make the 11th, 21st, 31st, and 41st links of brass, which 
would at once show on which side of the middle of the chain was the doubtful mark. 
This would be particularly useful in Mining Surveying. 

f This must not be confounded with the pieces of wire which have the same 
name, since one of them is shorter than the " link " used in calculation by half a ring 
or more, according to the way in which the chain is made. 



MAKING TEE MEASUREMENTS. 9 

To reduce links to feet, subtract from the number of links as 
many units as it contains hundreds ; multiply the remainder by 2 
and divide by 3. 

To reduce feet to links, add to the given number half of itself, 
and add one for each hundred (more exactly, for each ninety-nine) 
in the sum. 

The chain is liable to be lengthened by its rings being pulled 
open, and to be shortened by its links being bent. It should 
therefore be frequently tested by a carefully measured length of 
66 feet, set out by a standard measure on a flat surface, such as the 
top of a wall, or on smooth level ground between two stakes, their 
centers being marked by small nails. It may be left a little longer 
than the true length, since it can seldom be stretched so as to be 
perfectly horizontal and not hang in a curve, or be drawn out in 
a perfectly straight line.* Distances measured with a perfectly 
accurate chain will always and unavoidably be recorded as longer 
than they really are. To insure the chain being always strained 
with the same force, a spring, like that of a spring-balance, is 
sometimes placed between one handle and the rest of the chain. 

If a line has been measured with an incorrect chain, the true 
length of the line will be obtained by multiplying the number of 
chains and links in the measured distance by 100, and dividing by 
the length of the standard distance, as given by measurement of 
it with the incorrect chain. The proportion here employed is 
this : As the length of the standard given by the incorrect chain 
is to the true length of the standard, so is the length of the line 
given by the measurement to the true length. Thus, suppose that 
a line has been measured with a certain chain, and found by it to 
be ten chains long, and that the chain is afterward found to have 
been so stretched that the standard distance measured by it appears 
to be only 99 links long. The measured line is therefore longer 
than it had been thought to be, and its true length is obtained by 
multiplying 10 by 100, and dividing by 99. 



* The chain used by the Government surveyors of France, which is ten metres, 
or about half a Gunter's chain in length, is made from one fifth to two fifths of an 
inch longer than the standard. An inaccuracy of one five-hundredth of its length 
(= li inch on a Gunter's chain) is the utmost allowed not to vitiate the survey. 



10 LAND-SURVEYING. 

17. Pins. Ten iron pins, or " arrows," usually accompany the 
chain.* They are about afoot long, and are made of stout iron 
wire, sharpened at one end, and bent into a ring at the other. 
Pieces of red and white cloth should be tied to their heads, so that 
they can be easily found in grass, dead leaves, etc. 

They should be strung on a ring, which has a spring-catch to 

retain them. Their usual form is shown in Fig. 9. Fig. 10 shows 

another form, made very large, and therefore 

a very heavy near the point, so that, when held by 
the top and dropped, it may fall vertically. The 
uses of this will be seen presently. 

On irregular ground, two stout stakes, about 

six feet long, are needed to put the forward 

chain-man in line, and to enable whichever of 

the two is lowest to raise his end of the chain in 

y a truly vertical line, and to strain the chain 

straight. 

A number of long and slender rods are also necessary for 

"ranging out" lines between distant points. 

18. How to Chain. Two men are required — a forward chain- 
man and a hind chain-man, or leader and follower. The latter 
takes the handles of the chain in his left hand, and the chain itself 
in his right hand, and throws it out in the direction in which it is 
to be drawn. The former takes a handle of the chain and one pin 
in his right hand, and the other pins (and the staff, if used), in his 
left hand, and draws out the chain. The follower then walks be- 
side it, examining carefully that it is not twisted or bent. He 
then returns to its hinder end, which he holds at the beginning of 
the line to be measured, puts his eye exactly over it and, by the 
words "Eight," "Left," directs the leader how to put his staff, 
or the pin which he holds up, "in line," so that it may seem to 
cover and hide the flag-staff, or other object at the end of the line. 
The leader all the while keeps the chain tightly stretched, and his 

* Eleven pins are sometimes used, one being of brass. Nine of iron, with four 
or eight of brass, may also be employed. Their uses are explained in Articles 18 
and 19. 



MAKING THE MEASUREMENTS. 11 

end of it touching his staff. Every time he moves the chain, he 
should straighten it by an undulating shake. When the staff (or 
pin) is at last put "in line," the follower says "Down." The 
leader then puts in the single pin precisely at the end of the chain, 
and replies "Down." The follower then (and never before hear- 
ing this signal that the point is fixed) loosens his end of the chain, 
retaining it in his hand. The leader draws on the chain, making 
a step to one side of the pin just set, to avoid dragging it out. He 
should keep his eye steadily on the object ahead, or, in a hollow, 
should line himself approximately by looking back. The follower 
should count his steps, so as to know where to look for the pin in 
high grass, etc. As he approaches the pin, he calls "Halt." On 
reaching it, he holds the handle of the chain against it, pressing 
his knee against both to keep the pin firm. He then, with his eye 
over the pin, "lines "the leader as before. "When the "Down" 
has been again called by the follower, and answered by the leader, 
the former pulls out the pin with the chain-hand, and carries it in 
his other hand, and they go on as before.* The operation is re- 
peated till the leader has arrived at the end of the line, or has put 
down all his pins. 

When the leader has put down his tenth pin, he draws on the 
chain its length farther, and, after being lined, puts his foot on 
the handle to keep it firm, and calls " Tally." The follower then 
drops his end of the chain, goes up to the leader and gives him 
back all the pins, both counting them to make sure that none have 
been lost. One pin is then put down at the forward end of the 
chain, and they go on as before. 

Some surveyors cause the leader to call " tally " at the tenth 
pin, and then exchange pins ; but then the follower has only the 
hole made by the pin, or some other indefinite mark, to measure 
from. 

Eleven pins are sometimes preferred, the eleventh being of 
brass, or otherwise different from the rest, and being used to mark 

* When a chain's length would end in a ditch, pool of water, etc., and the chain- 
men are afraid of wetting their feet, they can measure part of a chain, to the edge of 
the water, then stretch the chain across it, and then measure another portion of a 
chain, so that, with the former portion, it may make up a full chain. 

2 



12 LAND-SURYEYING. 

the end of the eleventh chain ; another being substituted for it 
before the leader goes on. 

The two chain-men may change duties at each change of pins, 
if they are of equal skill, but the more careful and intelligent of 
two laborers should generally be made "follower." 

When the leader reaches the end of the line, he stops, and 
holds his end of the chain against it. The follower drops his end 
and counts the links beyond the last pin, noting carefully on which 
side of the " fifty " mark it comes. Each pin now held by the 
follower, including the one in the ground, represents one chain ; 
each time "tally" has been called, and the pins exchanged, repre- 
sents ten chains, and the links just counted make up the total 
distance. 

19. Tallies. In chaining very long distances, there is danger 
of miscounting the number of "tallies," or tens. To avoid mis- 
takes, pebbles, etc., may be changed from one pocket into another 
at each change of pins ; or bits of leather on a cord may be slipped 
from one side to the other ; or knots tied on a string ; but the best 
plan is the following : Instead of ten iron pins, use nine iron pins, 
and four, or eight, or ten pins of brass, or very much longer than 
the rest. At the end of the tenth chain, the iron pins being ex- 
hausted, a brass pin is put down by the leader. The follower then 
comes up, and returns the nine iron pins, but retains the brass 
one, with the additional advantage of having this pin to measure 
from. At the end of the twentieth chain, the same operation is 
repeated ; and so on. When the measurement of the line is com- 
pleted, each brass pin held by the follower counts ten chains, 
and each iron pin one, as before. 

20. Chaining on Slopes. All the distances employed in Land- 
surveying must be measured horizontally, or on a level. When 
the ground slopes, it is therefore necessary to make certain allow- 
ances or corrections. If the slope be gentle, hold the up-hill end 
of the chain on the ground, and raise the down-hill end till the 
chain is level. To insure the elevated end being exactly over the 
desired spot, raise it along a staff kept vertical, or drop a pin held 




MAKING THE MEASUREMENTS. 13 

by the point with the ring downward (if you have not the heavy 
pointed ones shown in Fig. 10), or, which is better, use a plumb- 
line. A person standing beside the chain, 
and at a little distance from it, can best tell 
if it be nearly level. If the hill be so steep 
that a whole chain can not be held up level, 
use only half or quarter of it at a time. 
Great care is necessary in this operation. 

To measure down a steep hill, stretch the whole chain in line. 
Hold the upper end fast on the ground. Eaise up the 20 or 30 . 
link-mark, so that that portion of the chain is level. Drop a 
plumb-line or pin. Then let the follower come forward and hold 
down that link on this spot, and the leader hold up another 
short portion, as before. Chaining down a slope is more ac- 
curate than chaining up it, since in the latter case the fol- 
lower can not easily place his end of the chain exactly over the 
pin. 

A more accurate, though more troublesome, method, is to 
measure the angle of the slope, and make the proper allowance 
by calculation, or by a table, previously prepared. The correc- 
tion being found, the chain may be drawn forward the proper 
number of links, and the correct distance of the various points 
to be noted will thus be obtained at once, without any subse- 
quent calculation or reduction. If the survey is made with 
the Transit provided with a vertical circle, the slope of the 
ground can be measured directly. A " Tangent Scale," for the 
same purpose, may be formed on the sides of the sights of 
a Compass. It will be described when the instrument is ex- 
plained. 

In the following table, the first column contains the angle 
which the surface of the ground makes with the horizon ; the 
second column contains its slope, named by the ratio of the per- 
pendicular to the base ; and the third, the correction in links for 
each chain measured on the slope, i. e., the difference between the 
hypothenuse, which is the distance measured, and the horizontal 
base, which is the distance desired. 



14 



LAND-SUR VEYING. 



TABLE FOE CHAINING ON SLOPES. 



Angle. 


Slope. 


Correction in 
links. 


Angle. 


Slope. 


Correction in 
links. 


3° 


1 in 19 


0-14 


13° 


1 in 4| 


2-56 


4° 


1 in 14 


0-24 


14° 


1 in 4 


2-97 


5° 


1 in 111 


0-38 


15° 


1 in 4 


3-41 


6° 


1 in 9i 


0'55 


16° 


1 in 3| 


3-87 


7° 


1 in 8 


0-75 


17° 


1 in 3£ 


4-37 


8° 


1 in 7 


0-97 


18° 


1 in Si 


4-89 


9° 


1 in 6| 


1-23 


19° 


1 in 3 


545 


10° 


1 in 6 


1-53 


20° 


1 in 2| 


6-03 


11° 


1 in 5£ 


1-84 


25° 


1 in 2 


9-37 


12° 


1 in 4f 


2'19 


30° 

1 


1 in If 


13-40 



21. Chaining is the fundamental operation in all kinds of Sur- 
veying. It has for this reason been very minutely detailed. The 
"follower" is the most responsible person, and the surveyor will 
best insure his accuracy by* taking that place himself. If he has 
to employ inexperienced laborers, he will do well to cause them to 
measure the distance between any two points, and then remeasure 
it in the opposite direction. The difference of their two results 
will impress on them the necessity of great carefulness. 

To " do up " the chain, take the middle of it in the left hand, 
and with the right hand take hold of the doubled chain just be- 
yond the second link ; double up the two links between your 
hands, and continue to fold up two double links at a time, laying 
each pair obliquely across the others, so that when it is all folded 
up the handles will be on the outside, and the chain will have an 
hour-glass shape, easy to strap up and to carry. 



22. Tape. Though the chain is most usually employed for the 
principal measurements of Surveying, a tape-line, divided on one 
side into links, and on the other into feet and inches, is more con- 
venient for some purposes. It should be tested very frequently, 
particularly after getting wet, and the correct length marked on it 
at every ten feet. A " Metallic Tape," less liable to stretch, is 
manufactured, in which fine wires form its warp. When the tape 
is being wound up, it should be passed between two fingers to pre- 
vent its twisting in the box, which would make it necessarv to 
unscrew its nut to take it out and untwist it. "While in use, it 



MAKING THE MEASUREMENTS. 15 

should be made portable by being folded up by arm's lengths, in- 
stead of being wound up. 

A " Steel Tape," made of a thin ribbon of steel, with the divis- 
ions and numbers etched on it, is one of the most accurate meas- 
uring instruments. Those intended for accurate measurement 
have at one end an arrangement for shortening and lengthening 
the tape to provide for variations in length, due to changes of tem- 
perature, and at the other end a level and a spring-balance, so that 
when measuring the ends of the tape may be held at the same 
height, and always with the same tension. For methods employed , 
in making accurate measurements, see Part IV. 

23. Substitutes for a chain or a tape may be found in leather 
driving-lines, marked off with a carpenter's rule, or in a cord knot- 
ted at the length of every link. A well-made rope (such as a 
" patent wove line," woven circularly with, the strands always 
straight in the line of the strain), when once well stretched, wetted, 
and allowed to dry with a moderate strain, will not vary from a 
chain more than one foot in two thousand, if carefully used. 

24. Rods. When unusually accurate measurements are re- 
quired, rods are employed. They may be of well-seasoned wood, 
of glass, of iron, etc. They must be placed in line very carefully 
end to end, or made to coincide in other ways, as will be explained 
under "Triangular Surveying," in which the peculiarly accurate 
measurement of one line is required, as all the others are founded 
upon it. 

25. Pacing, sound, and other approximate means, may be used 
for measuring the length of a line. The Stadia and Gradienter 
will be described in Chapter IV. 

26. A Perambulator, or " Measuring- Wheel," is sometimes used 
for measuring distances, particularly roads. It consists of a wheel 
which is made to roll over the ground to be measured, and whose 
motion is communicated to a series of toothed wheels within the 
machine. These wheels are so proportioned that the index-wheel 
registers their revolutions, and records the whole distance passed 



16 LAXD-SURVEYIXG. 

over. If the diameter of the wheel be 31-J- inches, the circumfer- 
ence, and therefore each revolution, will be 8J feet, or half a rod. 
The roughnesses of the road and the slopes necessarily cause the 
registered distances to exceed the true measure. 

The Odometer is an instrument designed to register the number 
of revolutions of a wagon-wheel. Knowing the circumference of 
the wheel to which it is attached, and determining the number of 
revolutions by the odometer, the distance over which the wheel 
has passed may be approximately determined. 

Measuring Angles, 

27. The angle made by any two lines — that is, the difference 
of their directions — may be obtained by a great variety of instru- 
Fm -[ 2 ments. All of them are in substance 

* mere modifications of the very sim- 
ple one which will now be described, 
and which any one can make for 
himself : 

Provide a circular piece of wood, 
and divide its circumference (by any 
of the methods of Geometrical Draft- 
ing) into three hundred and sixty 
equal parts, or " degrees," and num- 
ber them as in the figure. The divisions will be like those of a 
watch-face, but six times as many. These divisions are termed 
graduations. The figure shows only every fifteenth one. In the 
center of the circle fix a needle, or sharp-pointed wire, and upon 
this fix a straight stick, or thin ruler placed edgewise (called an 
alidade), so that it may turn freely on this point and nearly 
touch the graduations of the circle. Fasten the circle on a staff, 
pointed at the other end, and long enough to bring the alidade 
to the height of the eyes. The instrument is now complete. It 
may be called a Goniometer, or Angle-measurer. 

Now let it be required to measure the angle between the lines 
A B and A C. Fix the staff in the ground, so that its center shall 
be exactly over the intersection of the two lines. Turn the alidade 
so that it points (as determined by sighting along it) to a rod, or 




MAKING THE MEASUREMENTS. 17 

other mark at B, a point on one of the lines, and note what degree 
it covers — i. e., "The Beading." Then, without disturbing the 
circle, turn the alidade till it points to 0, a 
point on the other line. Note the new read- ' 

ing. The difference of these readings (in the / 

figure, 45 degrees) is the difference in the / \ 

directions of the two lines, or is the angle /*$t%\ 
which one makes with the other. If the dis- K£u£y 
tance from A to C be now measured, the 
point is " determined," with respect to the points A and B, on 
the Third Principle. Any number of points may be thus deter- 
mined. 

Instead of the very simple and rude alidade, which has been 
supposed to be used, needles may be fixed on each end of the ali- 
dade ; or sights may be added ; or a small straight tube 
' " may be used, one end being covered with a piece of 
W m7 pasteboard in which a very small eye-hole is pierced, 
and threads, called " cross-hairs," being stretched across 
the other end of it, as in the figure, so that their intersection may 
give a more precise line for determining the direction of any point. 

When a telescope is substituted for this tube, and supported in 
such a way that it can turn over, so as to look both backward and 
forward, the instrument (with various other additions, which, how- 
ever, do not affect the principle) is called a Transit. 

28. Chain Angles. The angle made by any two lines can also 
be determined without the aid of an angle-measurer. Let it be re- 
quired to find the angle made by the two lines 
A B and A C, Fig. 15. Measure off equal dis- 
tances from A to B and C, and also the '■' tie- 
line " B C. It is evident that the tie-line is the 
chord of the angle to a radius equal to one of the 
equal distances measured on the sides. Therefore, 
divide the length of the tie-line by the length 
of this distance. The quotient will be the chord of the angle to a 
radius of one. In the Table of Choeds, at the end of this volume, 
find this quotient, and the number of degrees and minutes corre- 




18 LA2W-8URYE7IN0. 

spending to it gives the angle required. Otherwise, since the chord 
of any angle equals twice the sine of half the angle, we have this 
rule : Divide half the tie-line by the measured distance, find in a 
table of natural sines the angle corresponding to the quotient, and 
multiply this angle by two, to get the angle desired. 

Surveying without Instruments. 

29. Distances by Pacing. Quite an accurate measurement of a 
line of ground may be made by walking over it at a uniform pace, 
and counting the steps taken. But the art of walking in a straight 
line must first be acquired. To do this, fix the eye on two objects 
in the desired line, such as two trees, or bushes, or stones, or tufts 
of grass. Walk forward, keeping the nearest of these objects stead- 
ily covering the other. Before getting up to the nearest object, 
choose a new one in line farther ahead, and then proceed as before, 
and so on. It is better not to attempt to make each of the paces 
three feet, but to take steps of the natural length, and to ascertain 
the value of each by walking over a known distance, and dividing 
it by the number of paces required to traverse it. Every person 
should thus determine the usual length of his own steps, repeating 
the experiment sufficiently often. The French "geographical en- 
gineers " accustom themselves to take regular steps of eight tenths 
of a metre, equal to two feet seven and a half inches. The United 
States military pace is two feet and six inches. This is regarded as 
a usual average. Quick pacing of 120 such paces per minute gives 
3*41 miles per hour. Slow paces, of three feet each and sixty per 
minute, give 2*04 miles per hour.* 

The Pedometer is an instrument which counts the steps taken 
by one wearing it, without any attention on his part. It is made 
in the form of a watch, and carried in the pocket. The number of 
the steps given by the pedometer, multiplied by the length of the 
step, will give approximately any distance walked over. In one 
form of this instrument the number of steps is registered on a dial 
up to 2,500. 

In another form the instrument is intended to be regulated ac- 

* A horse, on a walk, averages 330 feet per minute, on a trot 650, and on a com- 
mon gallop 1,040. For longer times, the difference in horses is more apparent. 



MAKING TEE MEASUREMENTS. 19 

cording to the length of step of the person carrying it, and then 
the distance is registered on the dial in miles. 

30. Distances by Visual Angles. Prepare a scale, by marking 
off on a pencil what length of it, when it is held off at arm's length, 
a man's height appears to cover at different distances (previously 

Fig. 16. 




~L 



measured with accuracy) of 100, 500, 1,000 feet, etc. To apply 
this, when a man is seen at any unknown distance, hold up the 
pencil at arm's length, making the top of it come in the line from 
the eye to his head, and placing the thumb-nail in the line from 
the eye to his feet, as in Fig. 16. The pencil having been previ- 
ously graduated by the method above explained, the portion of it 
now intercepted between these two lines will indicate the corre- 
sponding distance. 

If no previous scale have been prepared, and the distance of a 
man be required, take a foot-rule, or any measure minutely divided, 
hold it off at arm's length as before, and see how much a man's 
height covers. Then, knowing the distance from the eye to the 
rule, a statement by the rule of three (on the principle of similar 
triangles) will give the distance required. Suppose a man's height, 
of 70 inches, covers one inch of the rule. He is then seventy times 
as far from the eye as the rule, and, if its distance be two feet, that 
of the man is 140 feet. Instead of a man's height, that of an ordi- 
nary house, of an apple-tree, the length of a fence-rail, etc. , may 
be taken as the standard of comparison. 

To keep the arm immovable, tie a string of known length to 
the pencil, and hold between the teeth a knot tied at the other end 
of the string. 

31. Distances by Visibility. The degree of visibility of various 
well-known objects will indicate approximately how far distant they 



20 LAND-SURVEYING. 

are. Thus, by ordinary eyes, the windows of a large house can be 
counted at a distance of about 13,000 feet, or 2 J- miles ; men and 
horses will be perceived as points at about half that distance, or If 
mile ; a horse can be clearly distinguished at about 4,000 feet ; the 
movements of men at 2, 600 feet, or half a mile ; and the head of a 
man, occasionally, at 2,300 feet, and very plainly at 1,300 feet, or a 
quarter of a mile. The Arabs of Algeria define a mile as " the dis- 
tance at which you can no longer distinguish a man from a wom- 
an." These distances of visibility will of course vary somewhat 
with the state of the atmosphere, and still more with individual 
acuteness of sight, but each person should make a corresponding 
scale for himself. 

32. Distances by Sound. Sound passes through the air with a 
moderate and known velocity ; light passes almost instantaneously. 
If, then, two distant points be visible from each other, and a gun 
be fired at night from one of them, an observer at the other, noting 
by a stop-watch the time at which the flash is seen, and then that 
at which the report is heard, can tell by the intervening number of 
seconds how far apart the points are, knowing how far sound trav- 
els in a second. Sound moves about 1,098 feet per second in dry 
air, with the temperature at the freezing-point, 32° Fahr. For 
higher or lower temperatures add or subtract 1^- foot for each de- 
gree of Fahrenheit. If a wind blows with or against the movement 
of the sound, its velocity must be added or subtracted. If it blows 
obliquely, the correction will evidently equal its velocity multiplied 
by the cosine of the angle which the direction of the wind makes 
with the direction of the sound. If the gun be fired at each end of 
the base in turn, and the means of the times taken, the effect of 
the wind will be eliminated. 

If a watch is not at hand, suspend a pebble to a string (such as 
a thread drawn from a handkerchief) and count its vibrations. If 
it be 39-J inches long, it will vibrate in one second ; if 9f inches 
long, in half a second, etc. If its length is unknown at the time, 
still count its vibrations ; measure it subsequently ; and then will 

the time of its vibration, in seconds, = \/( — - — ^ry- -). 



DRAWING THE MAP. 21 

33. Angles. Right angles are those most frequently required in 
this kind of survey, and they can be estimated by the eye with 
much accuracy. If other angles are desired, they will be deter- 
mined by measuring equal distances along the lines which make the 
angle, and then the line, or chord, joining the ends of these dis- 
tances, thus forming chain-angles, explained in Article 28. 

Noting the Measurements. 

34. The measurements which have been made, whether of lines 
or of angles, require to be very carefully noted and recorded. 
Clearness and brevity are the points desired. Different methods of 
notation are required for each of the systems of surveying which 
are to be explained, and will therefore be given in their appropriate 
places. 

DRAWING THE MAP. 

35. A Map of a survey represents the lines which bound the 
surface surveyed, and the objects upon it, such as fences, roads, 
rivers, houses, woods, hills, etc., in their true relative dimensions 
and positions. It is a miniature copy of the field, farm, etc., as it 
would be seen by an eye moving over it ; or as it would appear, if, 
from every point of its irregular surface, plumb-lines were dropped 
to a level surface under it, forming what is called, in geometrical 
language, its horizontal projection. 

36. Platting. A plat of a survey is a skeleton, or outline map. 
It is a figure " similar " to the original, having all its angles equal 
and its sides proportional. Every inch on it represents a foot, a 
yard, a rod, a mile, or some other length, on the ground ; all the 
measured distances being diminished 7 

in exactly the same ratio. 

Platting is repeating on paper, 
to a smaller scale, the measurements 
which have been made on the ground. 

Its various operations may there- 
fore be reduced, in accordance with 

the principles established in this chapter, to two, viz. : draw- 
ing a straight line in a given direction and of a given length ; 




22 LAND-SURYEYING. 

and describing an arc of a circle with a radius whose length is also 
given. The only instruments absolutely necessary for this are a 
straight ruler and a pair of "dividers" or "compasses." Others, 
however, are often convenient, and will be now briefly noticed. 

37. Straight Lines. These are usually drawn by the aid of a 

straight-edged ruler. But to obtain a very long straight line upon 
paper, stretch a fine silk thread between any two distant points, 
and mark in its line various points near enough together to be 
afterward connected by a common ruler. The thread may also be 
blackened with burned cork and snapped on the paper, as a car- 
penter snaps his chalk-line ; but this is liable to inaccuracies, from 
not raising the line vertically. 

38. Arcs. The arcs of circles used in fixing the position of a 
point on paper are usually described with compasses, one leg of 
which carries a pencil-point. A convenient substitute is a strip of 
pasteboard, through one end of which a fine needle is thrust into 
the given center, and through a hole in which, at the desired dis- 
tance, a pencil-point is passed, and can thus describe a circle about 
the center, the pasteboard keeping it always at the proper distance. 
A string is a still readier, but less accurate, instrument. 

39. Parallels. The readiest mode of drawing parallel lines is 
by the aid of a triangular piece of wood and a ruler. Let A B 

Fi lg be the line to which a parallel is to 

be drawn, and C the point through 

which it must pass. Place one side 

of the triangle against the line, and 

place the ruler against another side of 

the triangle. Hold the ruler firm 

and immovable, and slide the triangle 

along it till the side of the triangle 

which had coincided with the given line passes through the given 

point. This side will then be parallel to that given line, and a 

line drawn by it will be the line required. 

Another easy method of drawing parallels is by means of a T- 




DRAWING TEE MAP. 



23 



Fig. 19. 




square, an instrument very valuable for many other purposes. It 

is nothing but a ruler let into a thicker piece of wood, very truly 

at right angles to it. For this use 

of it, one side of the cross-piece 

must be even or " flush " with the 

ruler. To use it, lay it on the 

paper so that one edge of the ruler 

coincides with the given line A B. 

Place another ruler against the 

cross-piece, hold it firm, and slide 

the T-square along till its edge 

passes through the given point 0, 

as shown by the lower part of the 

figure. Then draw by this edge the desired line parallel to the 

given line. 

40. Perpendiculars. These may be drawn by the various prob- 
lems given in Geometry, but more readily by a triangle which has 
one right angle. Place the longest side of the triangle on the 

given line, and place a ruler against a 
second side of the triangle. Hold the 
ruler fast, and turn the triangle so as 
to bring its third side against the ruler. 
Then will the long side be perpendicu- 
lar to the given line. By sliding the 
triangle along the ruler, it may be used 
to draw a perpendicular from any point 
of the line, or from any point to the line. 

41. Angles. These are most easily set out with an instrument 
called a Protractor. This is usually a semicircle of brass, as in the 
figure, with its semi-circumference divided into 180 equal parts, or 
degrees, and numbered in both directions. It is, in fact, a minia- 
ture of the instrument (or of half of it) with which the angles have 
been measured. To lay off any angle at any point of a straight 
line, place the protractor so that its straight side, the diameter of 
the semicircle, is on the given line, and the middle of this diam- 
eter, which is marked by a notch, is at the given point. With a 



Fig. 20. 




24 



LAN3-SUR TEYIXG. 



needle or sharp pencil make a mark on the paper at the required 
number of degrees, and draw a line from the mark to the given 
point. 

Sometimes the protractor has an arm turning on its center and 

Fig. 21. 




extending beyond its circumference, so that a line can be at once 
drawn by it when it is set to the desired angle. A Vernier scale is 
sometimes added to it to increase its precision. 

A Rectangular Protractor is sometimes used, the divisions of 
degrees being engraved aloag three edges of a plane scale. The 

Fig. 22. 



SQ X 3q 



~7~ 



±50 



SO v 60 TO S O 9O1 00UU 130 130 



JLiJL 



l arO 



140 ISO 110110100 9080 70 60 50 AO 



^SO 



J!R 



-; 



% 



semicircular one is preferable. The objection to the rectangular 
protractor is that the division corresponding to a degree is very 
unequal on different parts of the scale, being usually two or three 
times as great at its ends as at its middle. 

A Protractor embracing an entire circle, with arms carrying 
verniers, is also sometimes employed, for the sake of greater 
accuracv. 



BRA WING TIIE MAP. 25 

42. Drawing 1 to Scale. The operation of drawing on paper 
lines whose length shall be a half, a quarter, a tenth, or any other 
fraction of the lines measured on the ground, is called "Drawing 
to Scale." 

To set off on a line any given distance to any required scale, 
determine the number of chains or links which each division of the 
scale of equal parts shall represent. Divide the given distance by 
this number. The quotient will be the number of equal parts to 
be taken in the dividers and to be set off. 

For example, suppose the scale of equal parts to be a common 
carpenter's rule divided into inches and eighths. Let the given 
distance be twelve chains, which is to be drawn to a scale of two 
chains to an inch. Then six inches will be the distance to be set 
off. If the given distance had been twelve chains and seventy-five 
links, the distance to be set off would have been six inches and 
three eighths, since each eighth of an inch represents twenty-five 
links. 

If the desired scale were three chains to an inch, each eighth of 
an inch would represent 37J links ; and the distance of 1,275 links 
would be represented by thirty-four eighths of an inch, or 4} 
inches. 

A similar process will give the. correct length to be set off for 
any distance to any scale. 

If the scale used had been divided into inches and tenths, as is 
much the most convenient, the above distances would have become 
on the former scale 6 T W inches, or nearly 6 T 4 7 inches ; and on the 
latter scale 4^^- inches, coming midway between the second and 
third tenth of an inch. 

Conversely, to find the real length of a line drawn on paper 
to any known scale, reverse the preceding operation. Take 
the length of the line in the dividers, apply it to the scale, 
and count how many equal parts it includes. Multiply their 
number by the number of chains or links which each represents, 
and the product will be the desired length of the line on the 
ground. 

This operation and the preceding one are greatly facilitated by 
the use of the scales to be described in Art. 47, 



26 LAND-SURVEYING. 

43. Scales. The choice of the scale to which a plat should be 
drawn — that is, how many times smaller its lines shall be than 
those which have been measured on the ground — is determined by 
several considerations. The chief one is that it shall be just large 
enough to express clearly all the details which it is desirable to 
know. A Farm Survey would require its plat to show every field 
and building. A State Survey would show only the towns, rivers, 
and leading roads. The size of the paper at hand will also limit 
the scale to be adopted. If the content is to be calculated 
from the plat, that will forbid it to be less than 3 chains to 1 
inch. 

Scales are named in various ways. TJiey should always oe ex- 
pressed fractionally — i. e., they should be so named as to indicate 
what fractional part of the real line measured on the ground, the 
representative line drawn on the paper, actually is. When custom 
requires a different way of naming the scale, both should be given. 
It would be still better if the denominator could always be some 
power of 10, or at least some multiple of 2 and 5, such as -g-J-jj-, 
ToW ToTo> 2tVo> e ^ c - For convenience in printing, these may be 
written thus : 1 : 500, 1 : 1,000, 1 : 2,000, 1 : 2,500, etc. 

Plats of Farm Surveys are usually named as being so many 
chains to an inch. 

Maps of Surveys of States are generally named as being made 
to a scale of so many miles to an inch. 

Maps of Railroad Surveys are said to be so many feet to an 
inch, or so many inches to a mile. 

44. Farm Surveys. If these are of small extent, two chains to 
one inch (Which is = 2 x 6 * x 12 = j^ = 1 : 1,584) is convenient. 

A scale of one chain to one inch (1 : 792) is useful for plans of buildings. 
Three chains to oue inch (1 : 2,376) is suitable for larger farms. It is the 
scale prescribed by the English Tithe Commissioners for their first-class 
maps. 

In France, the Cadastre Surveys are lithographed on a scale about equiv- 
alent to this, being 1 : 2,500. The original plans are drawn to a scale of 
1 : 5,000. Plans for the division of property are made on the former scale. 
"When the district exceeds 3,000 acres, the scale is 1 : 10,000. When it ex- 
ceeds 7,500 acres, the scale is 1 : 20,000. A common scale in France for 
small surveys is 1 : 1,000, about \\ chain to 1 inch. 



DBA WING THE MAP. 27 

45. State Surveys. On these surveys smaller scales are necessarily 
employed. 

On the United States Coast and Geodetic Survey all the scales are ex- 
pressed fractionally and decimally. "The surveys are generally platted 
originally on a scale of one to ten or twenty thousand, but in some instances 
the scale is larger or smaller. 

" These original surveys are reduced for engraving and publication, and, 
when issued, are embraced in three general classes : 1, small harbor-charts ; 
2, charts of bays and sounds ; and, 3, the General Coast Charts. 

" The scales of the first class vary from 1 : 10,000 to 1 : 60,000, according 
to the nature of the harbor and the different objects to be represented. 

" Where there are many shoals, rocks, or other objects, as in Nantucket 
Harbor and Hell Gate, or where the importance of the harbor makes it neces- 
sary, a larger scale'of 1 : 5,000, 1 : 10,000, and 1 : 20,000 is used. But where, 
from the size of the harbor or its ease of access, a smaller one will point out 
every danger with sufficient exactness, the scales of 1 : 40,000 and 1 : 60,000 
are used, as in the case of New Bedford Harbor, Cat and Ship Island Har- 
bor, New Haven, etc. 

" The scale of the second class, in consequence of the large areas to be 
represented, is usually fixed at 1 : 80,000, as in the case of New York Bay, 
Delaware Bay and River. Preliminary charts, however, are issued of various 
scales from 1 : 80,000 to 1 : 200,000. 

" Of the third class, the scale is fixed at 1 : 400,000 for the General Chart 
of the Coast from Gay Head to Cape Henlopen, although considerations of 
the proximity and importance of points on the coast may change the scales 
of charts of other portions of our extended coast." 

The National Survey of Great Britain is called, from the corps employed 
on it, the " Ordnance Survey." 

The "Ordnance Survey" of the southern counties of England was 
platted on a scale of 2 inches to 1 mile (1 : 81,680), and reduced for publi- 
cation to that of 1 inch to a mile (1 : 63,360). The scale of 6 inches to a 
mile (1 : 10,560) was adopted for the northern counties of England and for 
the southern counties of Scotland. The same scale was employed for plat- 
ting and engraving in outline the " Ordnance Survey " of Ireland. But a 
map on a scale of 1 inch to 1 mile (1 : 63,360) is now published, the former 
scale rendering the maps too unwieldy and cumbrous for consultation. 

The Ordnance Survey of Scotland was at first platted on a scale of 6 
inches to 1 mile (1 : 10,560). That scale has since been abandoned, and it is 
now platted on a scale of 2 inches to 1 mile (1 : 31,680), and the general maps 
are made to only half that scale. 

The Ordnance Survey scale for the maps of London and other large towns 
is 5 feet to 1 mile (1 : 1,056), or 1£ chain to 1 inch. 

In the " Surveys under the Public Health Act " of England, the scale for 
the general plan is 2 feet to 1 mile (1 : 2,640) ; and for the detailed plan 10 
feet per mile (1 : 528), or f of a chain per inch. 

The Government Survey of France is platted to a scale of 1:20,000. 
Copies are made to 1 : 40,000 : and the maps are engraved to a scale of 
1 : 80,000, or about f of an inch to 1 mile. 



28 LAXD-SURVEYim. 

Oassini'a famous map of France was on a scale of 1 : 86,400. 
The French War Department employ the scales of 1 : 10,000, 1 : 20,000, 
1 : 40,000, and 1 : 80,000 for the topography of France. 

46. Railroad Surveys. For these the New York Eailroad Law of 1880 
directs the horizontal scale of maps which are to be filed in the State Engi- 
neer's Office to be 500 feet to T V of a foot (= 1 : 5,000), and vertical scale 
for profiles to be 100 feet to T V of a foot (= 1 : 1,000). 

For the New York Canal Maps a horizontal scale of 2 chains to 1 inch 
(1 : 1,584), and a vertical scale of 20 feet to 1 inch, are employed. 

The parliamentary " standing orders " prescribe the plans of railroads, 
prepared for parliamentary purposes, to be made on a scale of not less than 
4 inches to the mile (1 : 15,840) ; and the enlarged portions (as of gardens, 
court-yards, etc.) to be on a scale not smaller than 400 feet to the inch 
(1:4,800). Accordingly, the practice of English railway engineers is to 
draw the whole plan to a scale of 6 chains, or 393 feet to the inch (1 : 4,752), 
as being just within the parliamentary limits. 

In France, the engineers of " Bridges and Eoads " (Corps des Ponts et 
Chaussees) employ for the general plan of a road a scale of 1 : 5,000, and for 
appropriations, 1 : 500. 

In the United States Engineer Service the following plans are pre- 
scribed: General plans of buildings, 1 inch to 10 feet (1 : 120). 
Maps of grounds, with horizontal curves one foot apart, 1 inch to 50 feet 

(1 : 600). 
Topographical maps, one mile and a half square, 2 feet to 1 mile (1 : 2,640). 
Do., comprising three miles square, 1 foot to one mile (1 : 5.280). 
Do., between four and eight miles square, 6 inches to one mile (1 : 10,560). 
Do., comprising nine miles square, 4 inches to one mile (1 : 15,840). 
Maps not exceeding 24 miles square, 2 inches to one mile (1 : 31,680). 
Maps comprising 50 miles square, 1 inch to one mile (1 : 63,360). 
Maps comprising 100 miles square, £ inch to one mile (1 : 126,720). 
Surveys of roads, canals, etc., 1 inch to 50 feet (1 : 600). 

47. The most convenient scales of equal parts are those of box- 
wood, or ivory, which, have a fiducial or feather edge, along which 
they are divided, so that distances can be at once marked off from 
this edge, without requiring to be taken off with the dividers ; or 
the length of a given line can be at once read off. Box-wood is 
preferable to ivory, as much less liable to warp, or to vary in length 
with changes in the moisture in the air. 

The student can, however, make for himself platting-scales of 
drawing-paper, or Bristol board. Cut a straight strip of this mate- 
rial, about an inch wide. Draw a line through, its middle, and set 



DRAWING THE MAP. 



29 



off on it a number of equal parts, each representing a chain to the 
desired scale. Subdivide the left-hand division into ten equal 

Fig. 23. 



parts, each of which will therefore represent ten links to this scale. 
Through each point of division on the central line, draw (with the 
T-square) perpendiculars extending to the edges, and the scale is 
made. It explains itself. The above figure is a scale of 2 chains 
to 1 inch. On it the distance 220 links would extend between the 
arrow-heads above the line in the figure ; 560 links extend between 
the lower arrow-heads, etc. 

A paper scale has the great advantage of varying less from a 
plat which has been made by it, in consequence of changes in the 
weather, than any other. The mean of many trials showed the 
difference between such a scale and drawing-paper, when exposed 
alternately to the damp open atmosphere, and to the air of a warm 
dry room, to be equal to *005, while that between box-wood scales 
and the paper was *012, or nearly 2J times as much. The differ- 
ence with ivory would have been even greater. 

Some of the more usual platting-scales are here given in their 
actual dimensions. 

In these five figures, different methods of drawing the scales 



1 i i i I I i i 



Fig. 24. — Scale of 1 chain to 1 inch. 
O 1 



have been given, but each method may be applied to any scale. 
The first and second, being the most simple, are generally the best. 
In the third the subdivisions are made by a diagonal line : the dis- 

Fig. 25.— Scale of 2 chains to 1 inch. 



s» (IF 



3 



tances between the various pairs of arrow-heads, beginning with the 
uppermost, are respectively 310, 540, and 270 links. 



30 



LAND-SUR VETING. 



In the fourth figure, the distances between the arrow-heads are 
respectively 310, 270, and 540 links. 



4 


) 


Fig. 26.- 

z. a 


—Scale 

4 


of 3 chains to 1 inch. 
i . 5 6 i 


r 8 < 


i 


"*■ 


_J _^ 










i 


) 


J. 




















f 


















i 


Jb 










, 






i 


\ 


f 










' 






i 




Jo 


















1 




N 














ski 




/ 












/ 


yo 


! 






1 


i ) 


1 1 




\ 



In the fifth figure, the scale of 5 chains to 1 inch is subdivided 
diagonally to only every quarter-chain, or 25 links. The distance 



Fig. 26 1 . — Scale of 4 chains to 1 inch. 
O 1 2 3 4 5 6 7 8 9 10 11 12 13 


\ t 










1 


i 


i i 


- 


\ /> 




y 










1 i 


\77 ! 


\l 






l - 






■ i 


< 


T// ' 


A 




. \ 


i ■ 






i ; 




5VO ! 


1 




■ 1 






) 



between the upper pair of arrow-heads on it is 12 J chains, or 12*25 ; 
between the next pair of arrow-heads it is 6 '50; and between the 
lower pair 14 '75. 



Fig. 27. — Scale of 5 chains to 1 inch. 



10 



J-JL 



A- 



A diagonal scale for dividing an inch, or half an inch, into 100 
equal parts, is found on the "plain scale" in every case of instru- 
ments. 



48. Vernier Scale. This is an ingenious substitute for the diag- 
onal scale. The one given in the following figure divides an inch 
into 100 equal parts, and, if each inch be supposed to represent a 
chain, it gives single links. 

Make a scale of an inch divided into tenths, as in the upper 
scale of the above figure. Take in the dividers eleven of these 
divisions, and set off this distance from the of the scale to the 



DRAWING TEE MAP. 



31 



left of it. Divide the distance thus set off into 10 equal parts. 
Each of them will be one tenth of eleven tenths of one inch, i. e., 



Fig. 28. 





100 5|0 <> 1()0 2{ i )(> 




















II 


II 1 




















1 

1 






J 


( 










' 


1 






' I 


88 66 ec 46- 2.2 

00 





































eleven hundredths, or a tenth and a hundredth, and the first di- 
vision on the short, or vernier scale, will overlap, or be longer than 
the first division on the long scale, by just one hundredth of an 
inch ; the second division will overlap two hundredths, and so on. 
The principle will be more fully developed in treating of " Ver- 
niers." 

Now, suppose we wish to take off from this scale 275 hundredths 
of an inch. To get the last figure, we must take five divisions on 
the lower scale, which will be 55 hundredths, for the reason just 
given ; 220 will remain, which are to be taken from the upper 
scale, and the entire number will be obtained at once by extending 
the dividers between the arrow-heads in the figure from 220 on the 
upper scale (measuring along its lower side) to 55 on the lower 
scale ; 254 would extend from 210 on the upper scale to 44 on the 
lower ; 318 would extend from 230 on the upper scale to 88 on the 
lower. Always begin then with subtracting 11 times the last figure 
from the given number ; find the remainders on the upper scale, 
and the number subtracted on the lower scale. 



49. A plat is sometimes made by a nominally reduced scale in 
the following manner : Suppose that the scale of the plat is to be 
ten chains to one inch, and that a diagonal scale of inches, divided 
into tenths and hundredths, is the only one at hand. By dividing 
all the distances by ten, this scale can then be used without any 
further reduction. But if the content is measured from the plat 
to the same scale, in the manner explained in the next chapter, the 
result must be multiplied by 10 times 10. This is called by old 
surveyors "raising the scale," or "restoring true measure " 



32 



LAXD-SUR YEYIXG. 



50. Sectoral Scales. The Sector (called by the French " Com- 
pass of Proportion'') is an instrument sometimes convenient for 
obtaining a scale of equal parts. It is in two portions, turning on 
a hinge, like a carpenter's pocket-rule. It contains a great num- 
ber of scales, but the one intended for this use is lettered at its 

ends L in English instru- 
Fm. 29. . , 

merits, and consists of two 

lines running from the center 
to the ends of the scale, and 
each divided into ten equal 
parts, each of which is again 
subdivided into ten, so that 
each leg of the scale contains 
100 equal parts. To illustrate 
its use, suppose that a scale 
of 7 chains to 1 inch is re- 
quired. Take 1 inch in the dividers, and open the sector till 
this distance will just reach from the 7 on one leg to the 7 on the 
other. The sector is then "set" for this scale, and the angle of 
its opening must not be again changed. Xow let a distance of 
580 links be required. Open the dividers till they reach from 58 
to 58 on the two legs, as in the dotted line in the figure, and it 
is the required distance. Again, suppose that a scale of 2-J 
chains to 1 inch is desired. Open the sector so that 1 inch shall 
extend from 25 to 25. Any other scale may be obtained in the 
same manner. 

Conversely, the length of any known line to any desired scale 
can thus be readily determined. 




51. Whatever scale may be adopted for platting the survey, it 
should be drawn on the map, both for convenience of reference 
and in order that the contraction and expansion caused by changes 
in the quantity of moisture in the atmosphere may affect the scale 
and the map alike. When the drawing-paper has been wet and 
glued to a board, and cut off when the map is completed, its con- 
tractions have been found by many observations to average from 
one fourth to one half per cent on a scale of 3 chains to an inch 



CALCULATING THE CONTENT. 33 

(1 : 2,376), which would therefore require an allowance of from one 
half perch to one perch per acre. 

A scale made as directed in Art. 47, if used to make a plat on 
unstretched paper, and then kept with the plat, will answer nearly 
the same purpose. 

Such a scale may be attached to a map by slipping it through 
two or three cuts in the lower part of the sheet, and will be a very 
convenient substitute for a pair of dividers in measuring any dis- 
tance upon it. 

52. Scale omitted. It may be required to find the unknown 
scale to which a given map has been drawn, its superficial content 
being known. Assume any convenient scale, measure the lines of 
the map by it, and find the content by the methods to be given in 
the next chapter, proceeding as if the assumed scale were the true 
one. Then make this proportion, founded on the geometrical 
principle that the areas of similar figures are as the squares of their 
corresponding sides : As the content found is to the given content, 
so is the square of the assumed scale to the square of the true scale. 

CALCULATING THE CONTENT. 

53. The Coktekt of a piece of ground is its superficial area, 
or the number of square feet, yards, acres, or miles which it con- 
tains. 

54. Horizontal Measurement. All ground, however inclined or 
uneven its surface may be, should be measured horizontally, or as 
if brought down to a horizontal plane, so that the surface of a hill, 
thus measured, would give the same content as the level base on 
which it may be supposed to stand, or as the figure which would 
be formed on a level surface beneath it by dropping plumb-lines 
from every point of it. 

This method of procedure is required for both geometrical and 
social reasons. 

Geometrically, it is plain that this horizontal measurement is 
absolutely necessary for the purpose of obtaining a correct plat. 
In Pig. 30, let A B C D and B E F be two square lots of ground, 



34 



LAXD-SUR VEYING. 



Fig. 30. 
B 




Fig. 32. 



Fig. 33. 



platted horizontally. Suppose the ground to slope in all directions 
from the point C, which is the summit of a hill. Then the lines 

B C, D C, measured on the slope, are 
longer than if measured on a level, 
and the field A BCD, of Fig. 30, 
platted with these long lines, would 
take the shapeABGD in Fig. 31; 
and the field B C E F, of Fig. 30, 
would become B H E F, of Fio*. 31. 
The two adjoining fields would thus 
overlap each other ; and the same 
difficulty would occur in every case 
of platting any two adjoining fields 
by the measurements made on the 
slope. 

Let us suppose another case, more simple than would ever occur 
in practice, that of a three-sided field, of equal sides, and composed 
of three portions, each sloping 
down uniformly (at the rate of 
one to one) from one point in 
the center, as in Fig. 32. Each 
slope being accurately platted, the 
three could not come together, but 
would be separated as in Fig. 33. 

TVe have here taken the most simple cases, those of uniform 
slopes. But with the common irregularities of uneven ground, 
to measure its actual surface would not only be improper, but im- 
possible. 

In the social aspect of this question, the horizontal measure- 
ment is justified by the fact that no more houses can be built on a 
hill than could be built on its flat base : 
and that no more trees, corn, or other 
plants, which shoot up vertically, can grow 
on it ; as is represented by the vertical 
lines in the figure.* Even if a side-hill 

* This question is more than two thousand years old, for Polybius writes : " Some 
even of those who are employed in the administration of states, or placed at the head 




Fig. 34. 




CALCULATING THE CONTENT. 35 

should produce more of certain creeping plants, the increased 
difficulty in their cultivation might perhaps balance this. For 
this reason the surface of the soil thus measured is sometimes 
called the productive base of the ground. 

Again, a piece of land containing a hill and a hollow, if meas- 
ured on the surface, would giye a larger content than it would 
after the hollow had been filled up by the hill, while it would yet 
really be of greater value than before. 

Horizontal measurement is called the " Method of Cultella- 
tion," and superficial measurement the " Method of Development. " * 

An act of the State of New York prescribes that "the acre, for 
land-measure, shall be measured horizontally." 

55. Unit of Content. The Acre is the unit of land-measure- 
ment. It contains 4 Eoods. A Rood contains 40 Perches. A 
Perch is a square Rod ; otherwise called a Pole. A Rod is 5 J 
yards, or 16 \ feet. 

Hence, 1 Acre = 4 Roods = 160 Perches = 4,840 square yards 
= 43,560 square feet. 

One square mile = 5,280 X 5,280 feet = 640 acres. 

Since a chain is 66 feet long, a square chain contains 4,356 
square feet ; and, consequently, ten square chains make one acre, f 

The French units of land-measure are the Are = 100 square 
Metres = 0*0247 acre = one fortieth of an acre, nearly ; and the 
Hectare = 100 Ares = 2*47 acres, or nearly two and a half. Their 
old land-measures were the " Arpent of Paris," containing 36,800 
square feet ; and the " Arpent of Waters and Woods," containing 
55,000 square feet. 

56. When the content of a piece of land (obtained by any of the 
methods to be explained presently) is given in square links, as is 

of armies, imagine that unequal and hilly ground will contain more houses than a 
surface which is flat and level. This, however, is not the truth. For, the houses, be- 
ing raised in a vertical line, form right angles, not with the declivity of the ground, but 
with the flat surface which lies below, and upon which the hills themselves also stand." 

* The former from cultellum, a knife, as if the hills were sliced off ; the latter so 
named because it strips off or unfolds, as it were, the surface. 

f Let the young student beware of confounding 10 square chains with 10 chains 
square. The former make one acre ; the latter space contains ten acres. 



36 LAXD-SURYEYINQ. 

customary, cut o£E four figures ou the right (i. e., divide by 10,000) 
to get it into square chains and decimal parts of a chain ; cut off 
the right-hand figure of the square chains, and the remaining fig- 
ures will be Acres. Multiply the remainder by 4, and the figure, if 
any, outside of the new decimal-point will be Roods. Multiply the 
remainder by 40, and the outside figures will be Perches. The 
nearest round number is usually taken for the Perches ; fractions 
less than a half -perch being disregarded.* 

Thus, 86*22 square chains = 8 Acres 2 Eoods 20 Perches. 

Also, 64-1818 do. = 6 A. 1 E. 27 P. 

43-7564 do. = 4 A. 1 E. 20 P. 

57. Chain Correction. "When a surrey has been made, and the 
plat has been drawn, and the content calculated ; and afterward 
the chain is found to have been incorrect, too short or too long, 
the true content of the land may be found by this proportion : 
As the square of the length of the standard given by the incorrect 
chain is to the square of the true length of the standard, so is the 
calculated content to the true content. Thus, suppose that the 
chain used had been so stretched that the standard distance meas- 
ured by it appears to be only 99 links long ; and that a square field 
had been measured by it, each side containing 10 of these long 
chains, and that it had been so platted. This plat, and therefore 
the content calculated from it, will be smaller than it should be, 
and the correct content will be found by the proportion 99 s : 
100 2 : : 100 square chains : 102*03 square chains. If the chain had 
been stretched so as to be 101 true links long, as found by com- 
paring it with a correct chain, the content would be given by this 
proportion : 100 2 : 101 2 : : 100 square chains : 102-01 square chains. 
In the former case, the elongation of the chain was l^g- true links ; 
and 100 3 : (101 ¥ V) 2 * * 100 square chains : 102*03 square chains. 

58. Boundary-Lines. The lines which are to be considered as 
bounding the land to be surveyed are often very uncertain, unless 
specified by the title-deeds. 

* To reduce square yards to acres, instead of dividing by 4,8-iO, it is easier, and 
very nearly correct, to multiply by 2, cut off four figures, and add to this product one 
third of one tenth of itself. 



CALCULATING THE CONTENT. 37 

If the boundary be a brook, the middle of it is usually the 
boundary-line. On tide-waters, the land is usually considered to 
extend to low-water mark. 

Where hedges and ditches are the boundaries of fields, as is 
almost universally the case in England, the dividing line is gen- 
erally the top edge of the ditch farthest from the hedge, both 
hedge and ditch belonging to the field on the hedge side. This 
varies, however, with the customs of the locality. From three to 
six feet from the roots of the quick- wood of the hedges are allowed 
for the ditches. 

Methods of Calculation. 

59. The various methods employed in calculating the content 
of a piece of ground may be reduced to four, which may be called 
Arithmetical, Geometrical, Instrumental, and Trigonometrical. 

60. First Method.— Arithmetically. From direct measure- 
ments of the necessary lines on the ground. 

The figures to be calculated by this method may be either the 
shapes of the fields which are measured, or those into which the 
fields can be divided by measuring various lines across them. 

The familiar rules of mensuration for the principal figures 
which occur in practice will be now briefly enunciated. 

61. Rectangles. If the piece of ground be rectangular in shape, its con- 
tent is found by multiplying its leugth by its breadth. 

62. Triangles. When the given quantities are one side of a triangle 
and the perpendicular distance to it from the opposite angle, the content of 
the triangle is equal to half the product of the side and the perpendicular. 

When the given quantities are the three sides of the triangle, add to- 
gether the three sides and divide the sum by 2 ; from this half sum sub- 
tract each of the three sides in turn ; multiply together the half sum and the 
three remainders ; take the square root of the product ; it is the content re- 
quired. If the sides of the triangle be designated by a, &, c, and their sum 
by s, this rule will give its area = J [|s (is — a) 
Fig. 86. (J. ^ ») (* ~ .)]. 

^s\ When two sides of a triangle and the included 

/^ \ \ . angle are given, its content equals half the prod- 

/^ \ uct of its sides into the siue of the included an- 

•^ p gle. Designating the angles of the triangle by 



38 LAFD-SUBYEYIKG. 

the capital letters A, B, C, and the sides opposite them by the corresponding 
small letters a, 5, c, the area = i o c sin. A. 

When one side of a triangle and the adjacent angles are given, its content 
equals the square of the given side multiplied by the sines of each of the 
given angles, and divided by twice the sine of the sum of these angles. Using 

1 1 U * i.1 2 Sm - B ' Sin ' - 

the same symbols as before, the area = a 1 —-. — ~ . 

J ' 2 sm. (B + C) 

When the three angles of a triangle and its altitude are given, its area, re- 
ferring to the above figure, = £ B D 2 . -= ^—. — ^ . 

sin. ^x . sm. \j 

63. Parallelograms, or four-sided figures whose opposite sides are par- 
allel. The content of a Parallelogram equals the product of one of its sides 
by the perpendicular distance between it and the side parallel to it. 

64. Trapezoids, or four-sided figures, two opposite sides of which are 
parallel. The content of a Trapezoid equals half the product of the sum of 
the parallel sides by the perpendicular distance between them. 

If the given quantities are the four sides a, &, c, d, of which o and d are 
parallel ; then, making q = $■ (a + o + c — d), the area of the trapezoid 

wil1 = v^i^ [q &-«)&-«)<*--* + *)]• 

When two parallel sides, o and d, and a third side, a, are given, and also 
the angle (7, which this third side makes with one of the parallel sides, then 

the content of the trapezoid = . a . sin. C. 



65. Trapeziums ; four-sided figures, none of whose sides are parallel. 

A very gross error, often committed as to this figure, is to take the aver- 
age, or half sum of its opposite sides, and multiply them together for the 
area: thus, assuming the trapezium to be equivalent to a rectangle with 
these averages for sides. 

In practical surveying, it is usual to measure a line across it from corner 
to corner, thus dividing it into two triangles, whose sides are known, and 
which can therefore be calculated by Art. 62. 

When two opposite sides, and all the angles are given, take one side and 
its adjacent angles (or their supplements, when their sum exceeds 180°), con- 
sider them as belonging to a triangle, and find its area by the second formula 
in Art. 62. Do the same with the other side and its adjacent angles. The 
difference of the two areas will be the area of the quadrilateral. 

When three sides and their two included angles are given, multiply together 
the sine of one given angle and its adjacent sides. Do the same with the 
sine of the other given angle and its adjacent sides. Multiply together the 
two opposite sides and the sine of the supplement of the sum of the given 
angles. Add together the first two products, and .add also the last product, 
if the sum of the given angles is more than 180°, or subtract it if this sum be 
less, and take half the result. Calling the given sides />, q, r, and the angle 



CALCULATING TEE CONTENT. 



39 



between^? and q = A ; and the angle between q and r = B ; the area of the 
quadrilateral 

= i[p . q sin. A + q . r . sin. B ± p . r sin. (180° — A — B)]. 
When the four sides and the sum of any two opposite angles are given, pro- 
ceed thus : Take half the sum of the four given sides, and from it subtract 
each side in turn. Multiply together the four remainders, and reserve the 
product. Multiply together the four sides. Take half their product, and 
multiply it by the cosine of the given sum of the angles increased by unity. 
Regard the sign of the cosine. Subtract this product from the reserved 
product, and take the square root of the remainder. It will be the area of 
the quadrilateral. 

When the four sides and the angle of intersection of the diagonals of the 
quadrilateral are given, square each side ; add together the squares of the 
opposite sides ; take the difference of the two sums ; multiply it by the tan- 
gent of the angle of intersection, and divide by four. The quotient will be 
the area. 

When the diagonals of the quadrilateral and their included angle are 
given, multiply together the two diagonals and the sine of their included 
angle, and divide by two. The quotient will be the area. 



66. Second Method.— Geometrically. 

of the necessary lines upon the plat. 



From measurements 



67. Division into Triangles. The plat of a piece of ground 
having been drawn from the measurements made by any of the 
methods which will be hereafter explained, lines may be drawn 
upon the plat so as to divide it into a number of triangles. Four 
ways of doing this are shown in the figures, viz. : by drawing lines 

Fig. 36. 



Fig. 37. 



Fig. 38. 







from one corner to the other corners ; from a point in one of the 
sides to the corners ; from a point inside of the figure to the cor- 
ners ; and from various corners to other corners. The last method 
is usually the best. The lines ought to be drawn so as to make 
the triangles as nearly equilateral as possible. 

One side of each of these triangles, and the length of the per- 
pendicular let fall upon it, being then measured, the content of 



40 LAXD-SUBVEYim. 

these triangles can be at once obtained by multiplying their base bv 
their altitude, and dividing by two. 

The easiest method of getting the length of the perpendicular, without 
actually drawing it, is to set one point of the dividers at the angle from 
which a perpendicular is to be let fall, and to open and shut their legs till an 
arc described by the other point will just touch the opposite side. 

Otherwise, a platting scale may be placed so that the zero-point of its 
edge coincides with the angle, and one of its cross-lines coincides with the 
side to which a perpendicular is to be drawn. The length of the perpen- 
dicular can then at once be read off. 

The method of dividing the plat into triangles is the one most commonly 
employed by surveyors for obtaining the content of a survey, because of the 
simplicity of the calculations required. Its correctness, however, is depend- 
ent on the accuracy of the plat, and on its scale, which should be as large as 
possible. Three chains to an inch is the smallest scale allowed by the Eng- 
lish Tithe Commissioners for plats from which the content is to be de- 
termined. 

In calculating in this way the content of a farm, and also of its separate 
fields, the sum of the latter ought to equal the former. A difference of one 
three-hundredth (^) is considered allowable. 

Some surveyors measure the perpendiculars of the triangles by a scale 
half of that to which the plat is made. Thus, if the scale of the plat be two 
chains to the inch, the perpendiculars are measured with a scale of one chain 
to the inch. The product of the base by the perpendicular thus measured, 
gives the area of the triangle at once, without its requiring to be divided 
by two. 

Another way of attaining the same end, with less danger of mistakes, is to 
construct a new scale of equal parts, longer than those by which the plat was 
made in the ratio ^/2 : 1 ; or 1*414 : 1. When the base and perpendicular 
of a triangle are measured by this new scale, and then multiplied together, 
the product will be the content of the triangle, without any division by two, 
In this method there is the additional advantage of the greater size and con- 
sequent greater distinctness of the scale. 

When the measurement of a plat is made some time after it has been 
drawn, the paper will very probably have contracted or expanded so that 
the scale used will not exactly apply. In that case a correction is necessary. 
Measure very precisely the present length of some line on the plat, of known 
length originally. Then make this proportion : As the square of the present 
length of this line is to the square of its original length, so is the content 
obtained by the present measurement to the true content. 

68. Graphical Multiplication. Prepare ft strip of drawing-paper, of a 
width exactly equal to two chains on the scale of the plat; i. e., one inch 
wide, as in the figure, for a scale of two chains to one inch ; two thirds of an 
inch wide for a scale of three chains ; half an inch for four chains, and so on. 
Draw perpendicular lines across the paper at distances representing one tenth 



CALCULATING TEE CONTENT. 



41 



of a chain on the scale of the triangle to be measured, thus making a plat- 
ting scale. Apply it to the triangle so that one edge of the scale shall pass 
through one corner, A, of the triangle, and the other edge through another cor- 
ner, B ; and note very precisely what divisions of the scale are at these points. 
Then slide the scale in such a way that the points of the scale which had 
coincided with A and B shall always remain on the line B A produced, till 
the edge arrives at the point 0. Then will A' C — that is, the distance, or 

Fig. 40. 




\C 



'B 

number of divisions on the scale, from the point to which the division A on 
the scale has arrived, to the third corner of the triangle — express the area of 
the triangle A B in square chains. 

For, from C draw a parallel to A B, meeting the edge of the scale in C, 
and draw C B. Then the given triangle A B = A B C\ But the area of 
this last triangle — AC multiplied by half the width of the scale, i. e., 
= AO'xl=AC. But, because of the parallels, A' = A C, therefore 
the area of the given triangle A B = A' C ; i. e., it is equal in square chains 
to the number of linear chains read off from the scale. This ingenious opera- 
tion is due to M. Cousinery. 

69. Division into Trapezoids. A line may be drawn across the 
field, as in Fig. 41, and perpendiculars drawn to it. The field 



Fig. 41, 



Fig. 42. 





42 



LAND-SUE YEYING. 



will thus be divided into trapezoids (excepting a triangle at each 
end), and their content can be calculated by Art. 64. . 

Otherwise : a line may be drawn outside of the figure, and per- 
pendiculars to it be drawn from each angle. In that case the 
difference between the trapezoids formed by lines drawn to the 
outer angles of the figure, and those drawn to the inner angles, 
will be the content. 

70. Division into Squares. Two sets of parallel lines, at right angles 
to each other, one chain apart (to the scale of the plat) may be drawn over 

the plat, so as to divide it into 
Fig. 43. squares, as in the figure. The 

number of squares which fall with- 
in the plat represent so many 
square chains ; and the triangles 
and trapezoids which fall outside 
of these may then be calculated 
and added to the entire square 
chains which have been counted. 

Instead of drawing the parallel 
lines on the plat, tbey may better 
be drawn on a piece of transparent 
" tracing-paper," which is simply 
laid upon the plat, and the squares 
counted as before. The same pa- 
per will answer for any number of plats drawn to the same scale. This 
method is a valuable and easy check on the results of other calculations. 

To calculate the fractional parts, prepare a piece of tracing-paper, or 
glass, by drawing on it one square of the same size as a square of the plat, 
and subdividing it, by two sets of ten parallels at right angles to each other, 
into hundredths. This will measure the fractions remaining from the former 
measurement, as nearly as can be desired. 





>- 




















































^ 






















\ 


















\ 


\ 




















\ 










\ 




















\ 


N 


J 





Fig. 44. 



71. Division into Parallelograms. Draw a series of parallel lines 
across the plat at equal distances depending on the scale. Thus, for a plat 
made to a scale of 2 chains to 1 inch, the distance 
between the parallels should be 2i inches ; for a 
scale of 3 chains to 1 inch, li inch ; for a scale 
of 4 chains to 1 inch, -§- inch ; for a scale of 5 
chains to 1 inch, -fa inch ; and for any scale, make 
the distance between the parallels that fraction 
of an inch which would be expressed by 10 divided 
by the square of the number of chains to the inch. 
Then apply a common inch scale, divided on the 
edge into tenths, to these parallels ; and every inch 



/"^> 


/ 


1 1 


\\ 


\ 


\ \ ' 


^ h 


! v 


\y 



CALCULATING TEE CONTENT 



43 



in length of the spaces included between each pair of them will be an acre, 

and every tenth of an inch will be a square chain. 

For, calling the number of chains to the inch, = n, and making the width 

10 10 10 

between the parallels — inch, this width will represent — - x n = — chains ; 
w 2 n* n 

10 
and as the inch length represents n chains, their product, — x n = 10 square 

n 

chains = 1 acre. 

To measure the triangles at the ends of the strips between the parallels, 
prepare a piece of glass, or stout tracing-paper, of a width equal to the 
width between the parallels, and draw a line through its middle longitudi- 
nally. Apply it to the oblique line at the end of the space between two 
parallels, and it will bisect the line, and thus reduce the triangle to an equiva- 
lent rectangle, as at A in the figure. "When an angle occurs between two 
parallels, as at B in the figure, the fractional part may be measured by any 
of the preceding methods. 

A somewhat similar method is much used by some surveyors, particularly 
in Ireland — the plat being made on a scale of 5 chains to 1 inch, parallel lines 
being drawn on it, half an inch apart, and the distances along the parallels 
being measured by a scale, each large division of which is T 8 „ inch in length. 
Each division of this scale indicates an acre ; for it represents 4 chains, and 
the distance between the parallels is 2| chains. This scale is called the 
" Scale of Acres." 

72. Addition of Widths. When the lines of the plat are very irregu- 
larly curved, as in the figure, draw across it a number of equidistant lines, 
as near together as the case may 
seem to require. Take a straight- 
edged piece of paper, and apply one 
edge of it to the middle of the first 
space, and mark its length from one 
end; apply the same edge to the 
middle of the next space, bringing 
the mark just made to one end, and 

making another mark at the end of the additional length ; so go on, adding 
the length of each space to the previous ones. "When all have been thus 
measured, the total length, multiplied by the uniform width, will give the 
content. 



Fig. 45. 




73. Third Method.— Instrumentally. 
tain instrumental operations on the plat. 



By performing cer- 



74. Reduction of a many-sided figure to a single equivalent 
triangle. Any plane figure bounded by straight lines may be re- 
duced to a single triangle, which shall have the same content. 

This can be done by any instrument for drawing parallel lines. 
4 



44 



LAND-SUB VETING. 



Fig. 46. 




Fig. 47. 



Let the trapezium, or four-sided figure, shown in Fig. 46, be re- 
quired to be reduced to a single equivalent triangle. Produce one 

side of the figure, as 4 — 1. 
Draw a line from the first to the 
third angle of the figure. From 
the second angle draw a parallel 
to the line just drawn, cutting 
the produced side in a point 1'. 
From the point 1' draw a line to 
the third angle. A triangle (1' — 
3 — 4 in the figure) will thus be 
formed, which will be equivalent to the original trapezium. 

For, the triangle 1 — 2 — 3 taken away from the original figure 
is equivalent to the triangle 1' — 1 — 3 added to it ; because both 
these triangles have the same base and also the same altitude, since 
the vertices of both lie in the same line parallel to the base. 

The content of this 
final triangle can then 
be found by measur- 
ing its perpendicular, 
and taking half the 
product of this per- 
pendicular by thebase. 

Let the given figure 
have five sides, as in Fig. 

47. For brevity, the angles of the figure will be named as numbered in the 
engraviug. Produce 5 — 1. Join 1 — 3. From 2 draw a parallel to 1 — 3, 
cutting the produced base in 1'. Join 1' — 4. From 3 draw a parallel to it, 
cutting the base in 2'. Join 2' — 4. Then will the triangle 2' — 4 — 5 be 
equivalent to the five-sided figure 1 — 2 — 3 — 4 — 5, for similar reasons to 
those of the preceding case. 

Let the given figure be 1 — 2 — 3 — 4 — 5 — 6 — 7 — 8, as shown in 
Fig. 48. All the operations are shown by dotted lines, and the finally 
resulting triangle, 5' — 7 — 8, is equivalent to the original figure of eight 
sides. 

It is best, in choosing the side to be produced, to take one which has a 
long side adjoining it on the end not produced ; so that this long side may 
form one side of the final triangle, the base of which will therefore be 
shorter, and will not be cut so acutely by the final line drawn, as to make 
the point of intersection too indefinite. 




CALCULATING THE CONTENT 
Fig. 48. 



45 




75. General Rule. When the given figure has many sides, with 
angles sometimes salient and sometimes re-entering, the operations 
of reduction are very liable to errors if the draughtsman attempts 
to reason out each step. All difficulties, however, will be removed 
by the following General Rule : 

1. Produce one side of the figure, and call it a base. Call one 
of the angles at the base the first angle, and number the rest in 
regular succession around the figure. 

2. Draw a line from the 1st angle to the 3d angle. Draw a 
parallel to it from the 2d angle. Call the intersections of this 
parallel with the base the 1st mark. 

3. Draw a line from the 1st mark to the 4th angle. Draw a 
parallel to it from the 3d angle. Its intersection with the base is 
the 2d mark. 

4. Draw a line from the 2d mark to the 5th angle. Draw a 
parallel to it from the 4th angle. Its intersection with the base is 
the 3d mark. 

5. In general terms, which apply to every step after the first, 
draw a line from the last mark obtained to the angle whose number 
is greater by three than the number of the mark. Draw a parallel 
to it through the angle whose number is greater by two than that 
of the mark. Its intersection with the base will be a mark whose 
number is greater by one than that of the preceding mark. 

In the concise language of algebra, draw a line from the wth 



46 



LAXD-SUR YEYIXG. 



mark to the n + 3 angle. Draw a parallel to it through, the n-\-2 
angle, and the intersection with the base will be the n + 1 mark. 

6. Repeat this process for each angle, till you get a mark whose 
number is such that the angle haying a number greater by three is 
the last angle of the figure — i. e., the angle at the other end of the 
base. Then join the last mark to the angle which precedes the 
last angle in the figure, and the triangle thus formed will be the 
equivalent triangle required. 

In practice it is unnecessary to actually draw the lines joining 
the successive angles and marks, but the parallel ruler is merely 
laid on so as to pass through them, and the points where the par- 
allels cut the base are alone marked. 



Fig. 49. 



76. It is generally more convenient to reduce half of the figure on one 

side and half on the other, as is shown in 
Fig. 49, which represents the same field 
as Fig. 47. The equivalent triangle is 
herel'— 3 — 2'. 

TVhen the figure has many angles, they 
should not be numbered consecutively all 
the way around, but, after the numbers 
have gone around as far as the angle 
where it is intended to have the vertex 
of the final triangle, the numbers should 
be continued from the other angle of the 
base, as is shown in Fig. 50. In it only 
the intersections are marked. 
A figure with curved boundaries may be reduced to a triangle in a similar 
manner. Straight lines must be drawn about the figure, so as to be partly in 




Fig. 50. 



10 1 




CALCULATING TEE CONTENT. 



47 



it and partly out, giving and taking about equal quantities, so that the figure 
which these lines form shall be about equivalent to the curved figure. This 

Fig. 51. 




Fig. 52. 



having been done, tbe equivalent straight-lined figure is reduced by the above 
method. 

It is sometimes more convenient not to produce one of the 
sides of the figure, but to draw at one end of it, as at the point 
1 in Fig. 51, an indefinite line, usually a perpendicular, to a line 
joining two distant angles of the figure, and make this line the 
base of the equivalent triangle desired. The operation is shown 
by the dotted lines in the figure. The same General Eule ap- 
plies to it as to the previous figures. 

77. Special Instruments. A variety of instruments 
have been invented for the purpose of determining 
areas rapidly and correctly. 



One of the simplest is the " Computing Scale" which is on 
the same principles as the Method of Art. 71. It is repre- 
sented in Fig. 52. It consists of a scale divided for its whole 
length from the zero-point into divisions, each representing 
2£ chains to the scale of the plat. The scale carries a slider, 
which moves along it, and has a wire drawn across its center 
at right angles to the edges of the scale. On each side of this 
wire a portion of the slider, equal in length to one of the 
primary, or 2£ chain, divisions of the scale, is laid off and di- 
vided into 40 equal parts. 

This instrument is used in connection with a sheet of trans- 
parent paper, ruled with parallel lines at distances apart each 
equal to one chain on the scale of the plat. It is plain that 
when the instrument is laid on this paper, with its edge on one 
of the parallel lines, and the slider is moved over one of the 
divisions of 2£ chains, that one rood, or a quarter of an acre, 
has been measured between two of the parallel lines on the 
paper (since 10 square chains make one acre) ; and that one of 
the smaller divisions measures one perch between the same 
parallels. Four of the larger divisions give one acre. The scale 
is generally made long enough to measure at once five acres. 




48 



LAND-SUR V EYING. 





Fig. 
A 


53. 












/ 






> 


i i 


\ y 











To apply this to the plat of a field, or farm, lay the transparent paper 
over it in such a position that two of the ruled lines shall touch two of the 
exterior points of the boundaries, as at A and B. Lay the scale, with the 
slide set to zero, on the paper, in a direction parallel to the ruled lines, and 
so that the wire of the slide cuts the left-hand oblique line so as to make the 
spaces c aud d about equal. Hold the scale firm, and move the slider till the 
wire cuts the right-hand oblique line in such a way as to equalize the spaces 
e and/. Without changing the slide, move the scale down the width of a 

space and to the left-hand end of 
the next space ; begin there again, 
and proceed as before. 

So go on, till the whole length 
of the scale is run out (five acres 
having been measured), and then be- 
gin at the right-hand side and work 
backward to the left, reading the 
lower divisions, which run up to 10 
acres. By continuing this process, 
the content of plants of any size can 
be obtained. 

A still simpler substitute for this 
is a scale similarly divided, but with- 
out an attached slide. In place of 
it there is used a piece of glass, having a line drawn across it and riveted to 
the end of a short scale of box-wood, divided like the former slide. It is 
used like the former, except that, at starting, the zero of the short scale and 
not the line on the glass is made to coincide with the zero of the long 
scale. The slide is to be held fast to the instrument when this is moved. 

78. Planimeters. These determine the area of any figure, 
whether bounded by straight lines or curved, by merely moving 
a point around the outline of the surface. This causes motion in 
a train of wheel-work, which registers the algebraic sum of the 
product of ordinates to every point in that perimeter, by the incre- 
ment of their abscissas, and therefore measures the included space. 

There are several varieties of these instruments. One of the 
best of them is Amsler's Polar Planimeter. (For descriptions and 
theory of Planimeters, see " Mechanical Integrators/' by Henry 
S. H. Shaw.) 

79. A purely mechanical means of determining the area of any 
surface by means of its weight, may be placed here. The plat is 
cut out of paper and weighed by a delicate balance. The weight 
of a rectangular piece of the same paper containing just one acre 



CALCULATING TEE CONTENT. 49 

is also found ; and the " Bule of Three " gives the content. A 
modification of this is to paste a tracing of the plat on thin sheet- 
lead, cut out the lead to the proper lines and weigh it. 

80. Fourth Method.— Trigonometrically. By calculating, 
from the observed angles of the boundaries of the piece of ground, 
the lengths of the lines needed for calculating the content. 

This method is employed for surveys made with angular instru- 
ments, as the compass, etc., in order to obtain the content of the 
land surveyed, without the necessity of previously making a plat, 
thus avoiding both that trouble and the inaccuracy of any calcula- 
tions founded upon it. It is therefore the most accurate method ; 
but will be more appropriately explained in Part I, Chapter III, 
under the head of "Compass Surveying." 



CHAPTER II. 

CHAIN- SURVEYING ; BY THE FIRST AND SECOND METHODS I OR 
DIAGONAL AND PERPENDICULAR SURVEYING. 

81. The chain alone is abundantly sufficient, without the aid 
of any other instrument, for making an accurate survey of any 
surface, whatever its shape or size, particularly in a district tolera- 
bly level and clear. Moreover, since a chain, or some substitute 
for it, formed of a rope, of leather driving-reins, etc., can be ob- 
tained by any one in the most secluded place, this method of sur- 
veying deserves more attention than has usually been given to it. 

SURVEYING BY DIAGONALS : OR BY THE FIRST METHOD. 

82. Surveying by Diagonals is an application of the First 
Method of determining the position of a point, given in Art. 3, to 
which the student should again refer. Each corner of the field or 
farm which is to be surveyed is " determined " by measuring its 
distances from two other points. The field is then "platted" by 
repeating this process on paper, for each corner, in a contrary or- 
der, and the " content " is obtained by some of the methods ex- 
plained in Chapter I. 

The lines which are measured in order to determine the cor- 
ners of the field are usually sides and diagonals of the irregular 
polygon which is to be surveyed. They therefore divide it up into 
triangles ; whence this mode of surveying is sometimes called 
" Chain Triangulation." 

A few examples will make the principle and practice perfectly 
clear. Each will be seen to require the three operations of measur- 
ing, platting, and calculating. 



SURVEYING BY DIAGONALS. 51 

A three-sided field; as Fig. 54. 

Field- work. Measure the three sides, AB, BO, and C A. Measure also, 
as a proof-line, the distance from one of the corners, as C, to some point in 
the opposite side, as D, at which a mark should 
have been left, when measuring from A to B, Fig. 54. 

at a known distance from A. A stick or twig, . i 

with a slit in its top, to receive a piece of paper ^^ / j \ 

with the distance from A marked on it, is the /^ / i \ 

most convenient mark. ' A D is B 

Platting. Choose a suitable scale. Then 
draw a line equal in length, on the chosen scale, to one of the sides; AB, 
for example. Take in the compasses the length of another side, as A C, to 
the same scale, and with one leg in A as a center, describe an arc of a circle. 
Take the length of the third side, B 0, and, with B as a center, describe an- 
other arc, intersecting the first arc in a point which will be the third corner 
C. Draw the lines A and B ; and A B will be the plat, or miniature 
copy of the field surveyed. 

Instead of describing two arcs to get the point 0, two pairs of compasses 
may be conveniently used. Open them to the lengths, respectively, of the 
last two sides. Put one foot of each at the ends of the first side, and bring 
their other feet together, and their point of meeting will mark the desired 
third point of the triangle. 

To " prove " the accuracy of the work, fix the point D, by setting off from 
A the proper distance, and measure the length of the line D 0. If its length 
on the plat corresponds to its measurement on the ground, the work is cor- 
rect. 

It is a universal principle, in all surveying operations, that the work must 
be tested by some means independent of the original process, and that the 
same result must be arrived at by two different methods. The necessary 
length of this proof-line can also easily be calculated by the principles of 
trigonometry. 

Calculation. The content of the field may now be found, either from 
the three sides, or more easily though not so accurately, by measuring on 
the plat, the length of the perpendicular E, let fall from any angle to the 
opposite side, and taking half the product of these two lines. 

Example 1. Fig. 54 is the plat, on a scale of two chains to one inch, 
of a field, of which the side A B is 200 links, B is 100 links, and A is 150 
links. Its content, by the rule of Art. 62, is 0-726 of a square chain, or 
A. OR. 12 P. If the perpendicular OE be accurately measured, it will be 
found to be 72| links. Half the product of this perpendicular by the base 
will be found to give the same content. 

Ex. 2. The three sides of a triangular field are respectively 89*39, 54*08, 
and 45*98. Required its content. Ans. 100 A. R. 10 P. 

A four-sided field; as Fig. 55. 

Eield-worh Measure the four sides. Measure also a diagonal, as A O, 
thus dividing the four-sided field into two triangles. Measure also the other 
diagonal, or B D, for a " proof-line." 

Platting. Draw a line, as A C, equal in length to the diagonal, to any 



52 



LAND-SUB VETIXG. 




scale. On each side of it construct a triangle with the sides of the field, as 
directed above. 

To prove the accuracy of the work, measure on the plat the length of 

the "proof-line," BD, and if it 
Fig. 55. agrees with the length of the same 

C line measured on the ground, the 
field-work and platting are both 
proved to be correct. 

Calculation. Find the content 
of each triangle separately, as in 
the preceding case, and add them 
A- B together; or, more briefly, mul- 

tiply either diagonal (the longer 
one is preferable) by the sum of the two perpendiculars, and divide the prod- 
uct by two. 

Otherwise : reduce the four-sided figure to one triangle, as in Art. 74 ; or, 
use any of the methods of the preceding chapter. 

Ex. 3. In the field drawn in Fig. 55, on a scale of 3 chains to the inch, 
AB = 588 links, B C = 210, C D = 430, D A = 274, the diagonal A C= 626, 
and the proof diagonal B D = 500. The total content will be 1 A. R. IT P. 
Ex. 4. The sides of a four-sided field are A B = 12-41, B C = 5-86, C D 
= 8-25, DA = 4-24; the diagonal BD = 11-55, and the proof-line AC 
= 11-04. Required the content. Am. 4 A. 2 R. 38 P. 

Ex. 5. The sides of a four-sided field are as follows : A B = 8-95, B C 
= 5-33, CD = 10-10, D A= 6-54; the diagonal from A to C is 11-52; the 
proof diagonal from B to D is 10-92. Required the content. Arts. 

Ex. 6. In a four-sided field, A B = 7'68, B C = 4-09, CD = 10-64, D A 
= 7-24, A O = 10-32, B D = 10*74. Required the content. Am. 

A many-sided field, as Fig. 56. 
Field-work. Measure all the sides of the field. Measure also diagonals 




SURVEYING BY DIAGONALS. 53 

enough to divide the field into triangles, of which there will always bo two 
less than the number of sides. Choose such diagonals as will divide the 
field into triangles as nearly equilateral as possible. Measure also one or more 
diagonals for " proof -lines. 1 ' It is well for the surveyor himself to place 
stakes in advance at all the corners of the field, as he can then select the 
best mode of division. 

Platting. Begin with any diagonal and plat one triangle. Plat a sec- 
ond triangle adjoining the first one. Plat another adjacent triangle, and so 
proceed till all have been laid down in their proper places. Measure the 
proof-lines as before. 

Calculation. Proceed to calculate the content of the figure, precisely as 
directed for the four-sided field, measuring the perpendiculars and calculating 
the content of each triangle in turn ; or taking in pairs those on opposite 
sides of the same diagonal ; or using some of the other methods which have 
been explained. 

Ex. 7. The six-sided field, shown in Fig. 56, has the lengths of its lines, 
in chains and links, written upon them, and is divided into four triangles, 
by three diagonals. The diagonal B E is a " proof-line." The figure is drawn 
to a scale of 4 chains to the inch. The content of the field is 5 A. 3 E. 22 P. 

Ex. 8. In a five-sided field, the lengths of the sides are as follows : A B 
= 2-69, B = 1-22, C D = 2-32, D E = 3'55, E A = 3'23. The diagonals 
are A D = 4*81, BD = 3*33. Kequired its content. Ans. 

A field may be divided up into triangles, not only by measuring diagonals 
as in the last figure, but by any of the methods shown in the four figures of 
Art. 67. The one which we have been employing corresponds to the last of 
those figures. 

Still another mode may be used when the angles can not be seen from 
one another, or from any one point within. Take two or more convenient 
points within the field, and measure from them to the corners, and thus 
form different sets of triangles. 

Keeping the Field- Notes. 

83. By Sketch. The most simple method is to make a sketch 
of the field, as nearly correct as the unassisted hand and eye can 
produce, and note down on it the lengths of all the lines, as in 
Fig. 56. But when many other points require to be noted, such as 
where fences, or roads, or streams are crossed in the measurement, 
or any other additional particulars, the sketch would become con- 
fused, and be likely to lead to mistakes in the subsequent platting 
from it. The following is therefore the usual method of keeping 
the field-notes. A long, narrow book is most convenient for it. 

84. In Columns. Draw two parallel lines, about an inch apart 
from the bottom to the . top of the page of the field-book, as 



54 LAND-SUEYEYIXG. 

in the margin. This column, or pair of lines, may | 
be conceived to represent the measured line, split in 
two, its two halves being then separated, an inch 
apart, merely for convenience, so that the distances j 
measured along the line may be written between these | 
halves. 

Hold the book in the direction of the measure- . 
ment. At the bottom of the page write down the 
name, or number, or letter, which represents the station at which 
the survey is to begin. 



A " station " is marked with a triangle or circle, 
as in the margin. The latter is more easily made. 



A 
© 



562 



In the complicated cases, which will be hereafter explained, and 
in which one long base-line is measured; and also many other sub- 
ordinate lines, it will be well, as a help to the memory, to mark the 
stations on the base-line with a triangle, and the stations on the 
other lines with the ordinary circle. 

The station from which the measure- o to B 

ments are made is usually put on the left 
of the column ; and the station which is 
measured to, is put on the right. From A O 

But it is more compact, and avoids interfering 
with the notes of " offsets " (to be explained here- J* 
after), to write the name or number of the station in j^ 
the column, as in the margin. 

The measurements to different points of a line are B 
written above one another. The numbers all refer 050 
to the beginning of the line, and are counted from 100 

it. a 

The end of a measured line is marked by a line drawn across 
the page above the numbers which indicate the measurements 
which have been made. 

If the chaining does not continue along the adjoining line, but 
the chain-men go to some other part of the field to begin another 
measurement, two lines are drawn across the page. 



SURVEYING BY DIAGONALS. 



55 



When a line has been measured, the marks 
r or "] are made to show whether the following 
line turns to the right or to the left. 

A line is named, either by the names of the sta- 
tions between which it is measured, as the line 
A B ; or by its length, a line 562 links long, being 
called the line 562 ; or it is recorded as Line No. 1, 
Line No. 2, etc. ; or as Line on page 1, 2, etc., 
of the field-book. 

When a mark is left at any point of a line, as 
at D, in Fig. 49, with the intention of coming back 
to it again, in order to measure to some other point, 
the place marked is called a False Station, and 
is marked in the field-book "F. S."; or has a line 
drawn around it, to distinguish it ; or has a sta- 
tion mark A placed outside of the column, to the 
right or left, according to the direction in which 
the measurement from it is to be made. Examples 
of these three modes are given in the margin. 

A false station is named by its position on the 
line where it belongs ; as thus — " 200 on 562." 

When a gate occurs in a measured line, the distance from the 
beginning of the line to the side of the gate first reached is the 
one noted. 

When the measured line crosses a fence, brook, 
road, etc., they are drawn on the field-notes in 
their true direction, as nearly as possible, but not 
in a continuous line across the column, as in the 
first figure in the margin, but as in the second 
figure, so that the two parts would form a con- 
tinuous straight line, if the halves of the " split 
line " were brought together. 

It is convenient to name the lines in the mar- 
gin as being Sides, Diagonals, Proof-lines, etc. 





562 
200 

o 


F.S. 




562 

(200) 







562 

200 




A 




85. The field-notes of the triangular field platted in Fig. 54 are given 
below, according to both the methods mentioned in the preceding ar- 
ticle. 



56 



LAND-SUB VEYINO. 



In the field-notes in the column on the right hand, it is not absolutely 
necessary to repeat the B and C, 



From A 



200 

80 
O 





• 






89 


toG 


From D 


F.S. 






150 


to A 


From C 


O 


1 




100 


toG 


From B 


o 


1 



toB 
F.S. 



From 


C 
89 
( 80 ) \on 200 


1 


A 

150 

C 


1 


C 

100 
B 





B 

200 

® 

A 



86. The field-notes of the survey platted in Fig. 56 are given below. 
They begin at the bottom of the left-hand column. 





F 




IU 


532 




a 


300 


Gate. 




E 


r 





\ 


E 






\ 


662 


Brook. 


UJ 
Q 


\ 


400 




(/) 












D 


r 






D 








300 




ui 

Q 


— 


270 
210 

80 

O 


Boad* 


W 






S* 


r 





O 




ui 


703 




Q 


150 


Gate. 




B 


r 


ui 


B 




a 


562 




Ui 


A 





UJ 






z 
_l 


E 




u. 


770 




o 

O 


B 




tc 






a. 







\ 



E 

737 
280 

210 

A 







A 
1142 






1 











C 








775 


/ 




/ 


480 


Boad. 




/ 


420 






/ 








1 


E 





i 



oad. 





A 
270 
130 

80 

F 




Ui 
Q 


Boad. 




r 



SURVEYING BY TIE-LINES. 57 

SURVEYING BY TIE-LINES. 

87. Surveying dy Tie-lines is a modification of the method 
explained in the last chapter. It frequently happens that it is 
impossible to measure the diagonals of a field of many sides, in 
consequence of obstacles to measurements, such as woods, water, 
houses, etc. In such cases, " tie-lines " (so called because they 
tie the sides together) are employed as substitutes for diagonals. 

Thus, in the four-sided field shown in the Fig. 57. 

figure, the diagonals can not be measured 

because of woods intervening. As a substi- /jr>^c23?^ »■ 

tute, measure off from any convenient corner /<£*^Ov&p»/ 

of the field, as B, any distances, BE, BF, 

along the sides of the field. Measure also 

the " tie-line " E F. Measure all the sides of the field as usual. 

To plat this field, construct the triangle B E F, as in Art. 82. 
Produce the sides B E and B F, till they become respectively equal 
to B A and B C, as measured on the ground. Then, with A and 
as centers, and with radii respectively equal to A D and C D, 
describe arcs, whose intersection will be D, the remaining corner 
of the field. 

88. It thus appears that one tie-line is sufficient to determine a 
four-sided field, two a five-sided field, and so on. But, as a check 
on errors, it is better to measure a tie-line for each angle, and the 
agreement, in the plat, of all the measurements will prove the ac- 
curacy of the whole work. 

Since any inaccuracy in the length of a tie-line is increased in 
proportion to the greater length of the sides which it fixes, the tie- 
lines should be measured as far from the point of meeting of these 
sides as possible — that is, they should be as long as possible. 
The radical defect of the system is that it is "working from less 
to greater " (which is the exact converse of the 
m \J^\ true principle), thus magnifying inaccuracies at 

\i^S0}k ever J step * 

yf^^Ww A tie-line may also be employed as a " proof - 

\S2~^^/ line," in the place of a diagonal, and tested in 
C the same manner. 



58 



LAND-SUB YEYim. 



If any angle of the field is re-entering, as at B in the figure, 
measure a tie-line across the salient angle ABC. 

89. Chain-Angles. It is convenient, though not necessary, to 
measure equal distances along the sides : B E, B F, in Fig. 57, 
and B A, B C, in Fig. 58. " Chain-angles " are thus formed. To 
reduce "chain-angles" to degrees and minutes, see Art. 28. 




90. Inaccessible Areas. The method of tie-lines can be applied 
to measuring fields which can not be entered. 

Thus, in the figure, ABCD is an inaccessible wooded field, of 
four sides. To survey it, measure all the 
sides, and at any corner, as D, measure 
any distance D E, in the line of A D 
produced. Measure also another distance 
D F in the line of C D produced. Meas- 
ure the tie-line E F, and the figure can 
be platted as in the case of the field of 
Fig. 57, the sides of the triangle being produced in the contrary 
direction. 

The same end would be attained by prolonging only one side, as 
shown at the angle A of the same figure, and measuring A G, AH, 
and G H. It is better, in both cases, to tie all the angles in a 
similar manner. 

This method may be applied to a figure of any number of sides 
by prolonging as many of them as are necessary ; all of them, if 
possible. 



Fig. 60. 



91. If the sides C D and A D' were pro- 
longed by their full length, the content of 
the figure could be calculated without any plat ; 
for the new triangle D E F would equal the 
triangle D A C, and the sides of the triangle 
A C B would then be known. 

This principle may be extended still fur- 
ther. For a five-sided field, as in Fig. 60, pro- 
duce two pairs of sides, a distance equal to 





SURVEYING BY PERPENDICULARS. 59 

their length, forming two new triangles, as shown by the dotted 

lines, and measure the sides B'D' and A'D". The three sides of 

each of these triangles will thus be known, 

& ' Fig. 61. 

and also the three sides of the triangle BAD, ^..^ 

since AD=A' D", and B D = B' D'. 

The method of this article may be em- 
ployed for a figure of six sides, as shown in 
Fig. 61 (in which the dotted lines within 
the wooded field haye their lengths deter- 
mined by the triangles formed outside of it), 
but not for figures of a greater number of sides. 

SURVEYING BY PERPENDICULARS : OR BY THE SECOND 

METHOD. 

92. The method of Surveying hy Perpendiculars is founded 
on the Second Method of determining the position of a point, 
explained in Art. 4. It is applied in two ways, either to mak- 
ing a complete survey by "Diagonals and Perpendiculars," or to 
measuring a crooked boundary by "Offsets" Each will be con- 
sidered in turn. 

The best method of getting perpendiculars on the ground must, 
however, be first explained. 



Fig. 62. 




To set out Perpendiculars. 

93. Surveyor's Cross. The simplest instrument for 
this purpose is the Surveyor's Cross, or Cross-Staff, 
shown in the figure. It consists of a block of wood, 
of any shape, having in it two saw-cuts, made very 
precisely at right angles to each other, about half an 
inch deep, and with center-bit holes made at the bot- 
tom of the cuts to assist in finding the objects. This 
block is fixed on a pointed staff, on which it can turn 
freely, and which should be precisely 8 links (63J inches) 
long, for the convenience of short measurements. 

To use the cross-staff to erect a perpendicular, set 
it at the point of the line at which a perpendicular is 
wanted. Turn its head till, on looking through one 



60 LAND-SURVEYING. 

saw-cut, you see the ends of the line. Then will the other saw- 
cut point out the direction of the perpendicular, and thus guide 
the measurement desired. 

To find where a perpendicular to the line, from some object, as 
a corner of a field, a tree, etc., would meet the line, set up the 
cross-staff at a point of the line which seems to the eye to be about 
the spot. Note about how far from the object the perpendicular 
at this point strikes, and move the cross-staff that distance ; and 
repeat the operation till the correct spot is found. 

94. To test the accuracy of the instrument, sight through one 

slit to some point A, and place a stake B 

Fig. 63. R j n the line of sight of the other slit. 

I I / Then turn its head a quarter of the way 

\\J around, so that the second slit, looked 

A (t\ 4 through, points to A. Then see if the 

e$r other slit covers B again, as it will if 

correct. If it does not do so, but sights 

to some other point, as B', the apparent error is double the real 

one, for it now points as far to the right of the true point C as it 

did before to its left. 

This is the first example we have had of the invaluable prin- 
ciple of Reversion, which is used in almost every test of the accu- 
racy of surveying and astronomical instruments, its peculiar merit 
being that it doubles the real error, and thus makes it twice as easy 
to perceive and correct it. 

The instrument, in its most finished form, is made of a hollow 
brass cylinder, which has two pairs of slits exactly opposite to each 
other, one of each pair being narrow and the other wide, with a 
horse-hair stretched from the top to the bottom of the latter. It is 
also, sometimes, made with eight faces, and two more 
pairs of slits added, so as to set off half a right 
angle. 

Another form is a hollow brass sphere, as in the 
figure. This enables the surveyor to set off perpen- 
diculars on very steep slopes. 

Another form of the surveyor's cross consists of 




SURVEYING BY PERPENDICULARS. 



61 



Fig. 65. 



1 



Fig. 66. 




Fig. 67. 



two pairs of plain " Sights," each shaped as in the fig- 
ure, placed at the ends of two bars at right angles to 
each other. The slit, and the opening with a hair 
stretched from, its top to its bottom, are respectively at 
the top of one sight and at the bottom of the opposite 
sight.* This is used in the same manner as the preced- 
ing form, but is less portable, and more liable to get out of order. 
A temporary substitute for these instruments may 
be made by sticking four pins into the corners of a 
square piece of board, and sighting across them, in 
the direction of the line and at right angles to it. 

95. Optical Square. The most convenient and ac- 
curate instrument is, however, the Optical Square. The figures 
give a perspective view of it, and also a plan with the lid re- 
moved. It is a small circular box, 
containing a strip of looking-glass, 
from the upper half of which the 
silvering is removed. This glass is 
placed so as to make precisely half 
a right angle with the line of sight, 
which passes through a slit on one 
side of the box, and a vertical hair 
stretched across the opening on the 
other side, or a mark on the glass. 
The box is held in the hand over 
the spot where the perpendicular is 
desired (a plumb-line in the hand 
will give perfect accuracy), and the 
observer applies his eye to the slit 
A, looking through the upper or 
unsilvered part of the glass, and 
turns the box till he sees the other end of the line B, through the 
opening C. The assistant, with a rod, moves along in the direc- 
tion where the perpendicular is desired, being seen in the silvered 




* The French call the narrow opening ceillelon, and the wide one croisee. 



62 LAND-SURVEYING. 

parts of the glass, by reflection through the opening D, till his 
rod, at E, is seen to coincide with, or to be exactly under, the ob- 
ject B. Then is the line DE at right angles to the line A B, by 
the optical principle of the equality of the angles of incidence and 
reflection. 

To find where a perpendicular from a distant object would 
strike the line, walk along the line, with the instrument to the 
eye, till the image of the object is seen, in the silyered part of the 
glass, to coincide with the direction of the line seen through the 
unsilvered part. 

The instrument may be tested by sighting along the perpen- 
dicular, and fixing a point in the original line, on the principle of 
"reversion." 

The surveyor can make it for himself, fastening the glass in the 
box by four angular pieces of cork, and adjusting it by cutting 
away the cork on one side, and introducing wedges on the other 
side. The. box should be blackened inside. 

Another form of the optical square contains two glasses, fixed 
at an angle of 45°, and giving a right angle on the principle of the 
sextant. 

Perpendiculars may be set out with the chain alone, by a variety 
of methods. These methods generally consist in performing on 
the ground, the operations executed on paper in practical geome- 
try, the chain being used, in the place of the compasses, to describe 
the necessary arcs. 

As these operations, however, are less often used for the method 
of surveying now to be explained, than for overcoming obstacles to 
measurement, it will be more convenient to consider them in that 
connection. 

Diagonals and Perpendiculars. 

96. We have seen in the preceding pages that plats of surveys, 
made with the chain alone, have their contents most easily de- 
termined by measuring, on the plat, the perpendiculars of each of 
the triangles, into which the diagonals measured on the ground 
have divided the field. In the Method of Surveying lij Diagonals 
and Perpendiculars, now to be explained, the perpendiculars are 
measured on the ground. The content of the field can, therefore, 



SURVEYING BY PERPENDICULARS. 



63 



be found at once (by adding together the half products of each 
perpendicular by the diagonal on which it is let fall), without the 
necessity of previously making a plat, or of measuring the sides of 
the field. This is, therefore, the most rapid and easy method of 
surveying when the content alone is required, and is particularly 
applicable to the measurement of the ground occupied by crops, 
for the purpose of determining the number of bushels grown to the 
acre, the amount to be paid for mowing by the acre, etc. 



Fig. 68. 




A Three-sided Field. Measure the longest side, as A B, and the perpen- 
dicular, D, let fall on it from the opposite angle 
C. Then the content is equal to half the product 
of the side by the perpendicular. If obstacles pre- 
vent this, find the point, where a perpendicular let 
fall from an angle, as A, to the opposite side pro- 
duced, as B 0, would meet it, as at E in the figure. 
Then half the product of A E by C B is the content 
of the triangle. 

A Four-sided Field. Measure the diagonal A 0. Leave marks at the 
points on this diagonal at which perpendiculars from B and from D would 

meet it, finding these points by trial, 
as previously directed. The best marks 
at these " false stations " have been de- 
scribed. Return to these false stations 
and measure the perpendiculars. When 
these perpendiculars are measured be- 
fore finishing the measurement of the 
diagonal, great care is necessary to 
avoid making mistakes in the length of 
the diagonal, when the chainmen return 
to continue its measurement. One 
check is to leave at the mark as many 
pins as have been taken up by the hind-chainman in coming to that point 
from the beginning of the line. 

Ex. 9. Required the content of the field of Fig. 69. Am. A. 2 R. 29 P. 
The field may be platted from these measurements, if desired, but with 
more liability to inaccuracy than in the first method, in which the sides are 
measured. The plat of the figure is three chains to one inch. 

The field-notes may be taken by writing the measurements on a sketch, 
as in the figure ; or, in more complicated cases, by the column method, as 
below. A new symbol may be employed, this mark, f— , or H, to show the 
false station, from which a perpendicular is to be measured. 




u 



LA ND-SUR VFTIXG. 



£ From 200 on 480 



110 
F.S. 



to B 
H 



«=H From 280 <w 480 



175 
F.S. 



fo D 



iTz. 10. Calculation. 

sq. lies. 

ABC = \ x 480 x 110 = 26400 

ADC = \ x 480 x 175 = 42000 

sq. chains 6 "8400 

Acres 0'684 

It i3 still easier to take the two 
triangles together; multiplying the 
diagonal by the sum of the perpen- 
diculars and dividing hy two. 



A Many-sided Field. Fig. 70 and the accompanying field-notes repre- 
sent the field which was surveyed by the first method and platted in Fig. 56. 



_i 




480 


to C 


z 
o 




280 


H- 


(3 


H 


200 




a 


From A 


o 





From 5-07 on 7*37 



1-54 
F.S. 



to F 



2-53 toD 



From 1*60 on 7*75 


F.S. 




From 5-45 on 11-42 


4-93 
F.S. 


toE 


From 4' 95 on 11*42 


2*67 
F.S. 


toB 


From E 


7-37 

5-07 

O 


to A 

r 


From C 


7-75 
1-60 



r 



H 
From A 



11-42 

5-45 

4-95 

O 



toC 

»- 



JEfc. 11. Calculation. The con- 
tent of the triangles may be expressed 
thus : 

sq. Us. 

ABC = I- x 1142 x 267 = 152457 

AEC = \ x 1142 x 493 = 281503 

CDE = I x 775 x 253 = 98037 

AEF = £ x 737 x 154 = 56749 

sj. chains 58 - 8746 

Acres 5*88746 

or, 5 A. 3 R. 22 P. 

The first two triangles might have 
been taken together, as in the previous 
field. 

Content calculated from the per- 
pendiculars will generally vary slightly 
from that obtained by measuring on 
the plat. 



A small field which has many sides may sometimes be conveniently sur- 
veyed by taking one diagonal and measuring the perpendiculars let fall on it 
from each angle of the field, and thus dividing the whole area into triangles 
and trapezoids, as in Fig. 41. 

The line on which the perpendiculars are to be let fall may also be out- 
side of the field, as in Fig. 42. 

Such a survey can be platted very readily, but the length of the perpen- 
diculars renders the plat less accurate. 

This procedure supplies a transition to the method of " offsets." which is 
explained in the next article. 



SURVEYING BY PERPENDICULARS. 



65 




Offsets. 
97. Offsets are short perpendiculars, measured from a straight 
line, to the angles of a crooked or zigzag line near which the 
straight line runs. Thus, in the FlG ^ 1# 

figure, let A C D B be a crooked C 

fence, bounding one side of a field. 
Chain along the straight line A B, 
which runs from one end of the fence to the other, and, when 
opposite each corner, note the distance from the beginning, or the 
point A, and also measure and note the perpendicular distance of 

each corner and D from the line. 
These corners will then be "determined" 
by the Second Method, Art. 4. 

The field-notes, corresponding to Fig. 
71, are as in the margin. The measure- 
ments along the line are written in the 
column, as before, counting from the be- 
ginning of the line, and the offsets are 
written beside it, on the right or left, 
opposite the distance at which they are 
taken. A sketch of the crooked line is 
also usually made in the field-notes, though not absolutely neces- 
sary in so simple a case as this. The letters and D would not be 



D^25 
Cj 30 



Prom A 



300 



250 
100 







toB 



QQ 



LAND-SUE VEYING. 



used in practice, but are here inserted to show the connection 
between the field-notes and the plat. 

In taking the field-notes, the widths of the offsets should not 
be drawn proportionally to the distances between them, but the 
breadths should be greatly exaggerated in proportion to the 
lengths. 

A more extended example, with a little different notation, is 
given below. In the figure, which is on a scale of eight chains to 
one inch for the distances along the line, the breadths of the offsets 
are exaggerated to four times their true proportional dimensions. 



Fig. 72. 



: 





B 






1500 







1250 


20 





1000 





30 


750 




50 


500 




40 


250 

A 





Fig. 73. 



B 





250 


toB 


§0 30 


185 




30 20 


150 






90 


10 Fk 




50 


iol_J° 

30 


From A 


o 





aM 

The plat and field-notes of the position of two houses, deter- 
mined by offsets, are given above on a scale of two chains to one 
inch : 

Double offsets are sometimes convenient ; and sometimes triple 
and quadruple ones. Below are given the notes and the plat, one 
chain to one inch, of a road of varying width, both sides of which 
are determined by double offsets. It will be seen that the line A B 
crosses one side of the road at 160 links from A, and the other side 
of it at 220. 

Two methods of keeping the field-notes are given. In the first 
form, the offsets to each side of the road are given separately and 
connected by the sign +• In the second form, the total distance 
of the second offset is given, and the two measurements connected 
by the word "to." This is easier both for measuring and plat- 
ting. 



SURVEYING BY PERPENDICULARS. 



67 



Fig. 74. 




50 + 

55 + 
50 + 
45 + 
50 + 
50 + 
55 + 
60 + 





B 




260 




240 





220 


20 


200 


40 


180 


45 


160 





140 


5 


120 


20 


100 


15 


80 


10 


60 


20 


40 


20 


20 ; 





A 



30 4- 60 

10 + 70 
50 
30 
10 






B 




260 


• 


240 





220 


20 


200 


40 


180 


45 


160 


50 to 


140 


60 to 5 


120 


70 to 20 


100 


60 to 15 


80 


60 to 10 


60 


70 to 20 


40 


75 to 20 


20 


60 to 


A 



30 to 90 
10 to 80 
50 
30 
10 . 




Offsets may generally be taken with sufficient accuracy by 
measuring them as nearly at right angles to the base-line as the 
eye can estimate. The surveyor should stand by the chain, facing 
the fence, at the place which he thinks opposite to the corner to 
which he wishes to take an offset, and measure " square" to it by 
the eye, which a little practice will enable him to do with much 
correctness. 

The offsets may be measured, if short, with an Offset-staff, a 
light stick, 10 or 15 links in length, and divided accordingly ; or, 
if they are long, with a tape. They are generally but a few links 
in length. A chain's length should be the extreme limit, as laid 
down by the English "Tithe Commissioners," and that should be 
employed only in exceptional cases. When the " cross-staff " is in 
use, its divided length of 8 links renders the offset-staff needless. 

When offsets are to be taken, the method of chaining to the end 
of a line (described in Art. 18) is somewhat modified. After the 
leader arrives at the end of the line, he should draw on the chain 



68 LAND-SURVEYING. 

till the follower, with the back end of the chain, reaches the last 
pin set. This facilitates the counting of the links to the places 
at which the offsets are taken. 

The offsets are to be taken to every angle of the fence or other 
crooked line ; that is, to every point where it changes its direction. 
These angles or prominent bends can be best found by one of the 
party walking along the crooked fence and directing another at the 
chain what points to measure opposite to. If the line which is to 
be thus determined is curved, the offsets should be taken to points 
so near each other that the portions of the curved line lying be- 
tween them may, without much error, be regarded as straight. It 
will be most convenient, for the subsequent calculations, to take 
the offsets at equal distances apart along the straight line from 
which they are measured. 

In the case of a crooked brook, such as is shown in the figure 
given below, offsets should be taken to the most prominent angles, 
such as are marked a a am the figure, and the intermediate bends 
may be merely sketched by the eye. 

Fig. To. 



"^t 



When offsets from lines measured around a field are taken in- 
side of these bounding lines, they are sometimes distinguished as 

insets. 

98. Platting. The most rapid method of platting the offsets 
is by the use of a Platting Scale (described in Art. 47) and an Off- 
set Scale, which is a short scale divided on its edges like a platting 
scale, but having its zero in the middle, as in the figure. 

The platting scale is placed parallel to the line, with its zero- 
point opposite to the beginning of the line. The offset scale is 
slid along the platting scale, till its edge comes to a distance on 
the latter at which an offset had been taken, the length of which is 
marked off with a needle-point from the offset scale. This is then 



SURVEYING BY PERPENDICULARS. 



69 



slid on to the next distance, and the operation is repeated. If one 
person reads off the field-notes, and another plats, the operation 



Fig, 16. 




will be greatly facilitated. The points thus obtained are joined by 
straight lines, and a miniature copy of the curved line is thus ob- 
tained ; all the operations of the platting being merely repetitions 
of the measurements made on the ground. 

If no offset scale is at hand, make one of a strip of thick draw- 
ing-paper, or pasteboard ; or use the platting scale itself, turned 
crossways, having previously marked off from it the points from 
which the offsets had been taken. 

In plats made on a small scale, the shorter offsets are best esti- 
mated by the eye. 

On the ordnance survey of Ireland, the platting of offsets was 
facilitated by the use of a combination of the offset scale and the 
platting scale, the former being made to slide in a groove in the 
latter, at right angles to it. 



99. Calculating Content. When the crooked line determined by 
offsets is the boundary of a field, the content, inclosed between it 
and the straight line surveyed, must be determined, that it may 
be added to, or subtracted from, the content of the field bounded 
by the straight lines. There are various methods of effecting 
this. 

The area inclosed between the straight and the crooked lines is 
divided up by the offsets into triangles and trapezoids, the content 
of which may be calculated separately and then added together. 
The content of the plat on page 65 will, therefore, be 1500 + 
4125 -f 625 = 6250 square links = 0*625 square chain. The con- 



70 LAND-SURVEYING. 

tent of the plat on page 66 will in like manner be found to be, on 
the left of the straight line, 30,000 square links, and on its right, 
5,000 square links. 

100. Wlien the offsets have been taken at equal distances, the 
content may be more easily obtained by adding together half of 
the first and of the last offset, and all the intermediate ones, and 
multiplying the sum by one of the equal distances between the 
offsets. This rule is merely an abbreviation of the preceding 
one. 

Thus, in the plat of page 66, the distances being equal, the con- 
tent of the offsets on the left of the straight line will be 120 X 250 
= 30,000 square links, and on the right, 20 X 250 = 5,000 square 
links ; the same results as before. 

When the line determined by the offsets is a curved line, 
" Simpson's rule " gives the content more accurately. To employ 
it, an even number of equal distances must have been measured 
in the part to be calculated. Then add together the first and last 
offset, four times the sum of the even offsets (i. e., the 2d, 4th, 6th, 
etc.), and twice the sum of the odd offsets (i. e., the 3d, 5th, 7th, 
etc.), not including the first and the last. Multiply the sum by 
one of the equal distances between the offsets, and divide by 3. 
The quotient will be the area. 

Ex. 12. The offsets from a straight line to a curved fence 
were 8, 9, 11, 15, 16, 14, 9, links, at equal distances of 5 links. 
What was the content included between the curved fence and the 
straight line? Ans. 371-666. 

101. Equalizing, or giving and talcing, is an approximate mode 
of calculation much used by practical surveyors. A crooked line, 

Fig. nn. 



determined by offsets, having been platted, a straight line is drawn 
on the plat, across the crooked line, leaving as much space outside 



SURVEYING BY THE PRECEDING METHODS. 71 

of the straight line as inside of it, as nearly as can be estimated by 
the eye, "equalizing" it, or "giving and taking" equal portions. 
The straight line is best determined by laying across the irregular 
outline the straight edge of a piece of transparent horn, or tracing- 
paper, or glass, or a fine thread or horse-hair stretched straight by 
a light bow of whalebone. In practical hands, this method is suffi- 
ciently accurate in most cases. The student will do well to try it 
on figures, the content of which he has previously ascertained by 
perfectly accurate methods. 

SURVEYING BY THE PRECEDING METHODS COMBINED. 

102. All the methods which have been explained in the pre- 
ceding sections — surveying by Diagonals, by Tie-lines, and by 
Perpendiculars, particularly in the form of offsets — are frequently 
required in the same survey. The method by Diagonals should be 
the leading one ; in some parts of the survey obstacles to the meas- 
urements of diagonal may require the use of Tie-lines; and, if 
the fences are crooked, straight lines are to be measured near them, 
and their crooks determined by Offsets. 

Offsets are necessary additions to almost every other method of 
surveying. In the smallest field surveyed by diagonals, unless all 
the fences are perfectly straight lines, their bends must be deter- 
mined by offsets. The plat (scale of one chain to one inch) and 
field-notes of such a case are given below. A sufficient number of 
the sides, diagonals, and proof-lines, to prove the work, should be 
platted before platting the offsets. 

Fig. 18. 



72 



LAND-SUR YEYWG. 





c 







360 




6 


315 




10 


275 




5 


215 







150 







115 


10 




80 


5 




65 


8 




B 


or 




B 


| 





125 




11 


90 




23 


62 


1 


12 


22 


i 





A 





Ex. 13. Bequired the content 
of the above field. Ans. 



oz 

8^ 



B 

340 

D 





C 






310 






A 


r 




A 







248 




11 


180 







105 







65 


5 




D 


or 




D 







135 




15 


110 




13 


90 







50 







30 


9 




C 


or 



103. Field-Jooks. The difficulty and the importance of keep- 
ing the field-notes clearly and distinctly increase with each new 
combination of methods. For this reason, three different methods 
of keeping the field-notes of the same survey will now be given 
(from Bourn's "Surveying"), and. a careful comparison by the 
student of the corresponding portions of each will be very profit- 
able to him : 



SURVEYING BY THE PRECEDING METHODS. 



73 



Field-Book No. 1, 

Fig. 79, 



>6 




Field-Booh No. 1 (Fig. 79) shows the Sketch method, explained 
in Art. 83. 



74 



LAND-SUR VEYING. 



Field-Book No, 2. 

Fig. 80. 



\ 



.' 4570 ' 

4080 
(4000 ) 

('3650 ") 

3480 
3060 

('3020 ^i 

i 2450") 

V -' 

2300 

X 

,'"l300 N , 



1 1260 

X 



A 



760 
f'^SO 

1(00 

620 
260 

o 
A 



a 



toX 




Field-Boole No. 2 "(Fig. 80) shows the Column method, ex* 
plained in Art. 84. 



SURVEYING BY TEE PRECEDING METHODS. 



T5 



Field-Book No. 3. 

Fig. 81. 



5; *g> 





/ 


/ 


OSS 


/ 


1 1 


r 


~<f)£2 


0017 

0S9 






i 




0T8 


002/ 




fo^ei 






/ 5m x 







or / w 










o o 

O <N 

f CO 









Field-Booh No. 3 (Fig. 81) is a convenient combination of the 
two preceding methods. The bottom of the book is at the side of 
this figure, at A. 



76 



LAFD-SUR YEYING. 



104. Inaccessible Areas. 



Fig. 82. 




A combination of offsets and tie-lines 
supplies an easy method of 
surveying an inaccessible area, 
such as a pond, swamp, forest, 
block of houses, etc., as appears 
from the figure, in which exter- 
nal bounding lines are taken at 
will and measured, and tied 
by "tie-lines" measured be- 
tween these lines, prolonged 
when necessary, while offsets 
from them determine the irregularities of the actual boundaries of 
the pond, etc. 

These offsets are insets, and their content is, of course, to be 
subtracted from the content of the principal figure. 

Even a circular field might thus 
be approximately measured from the 
outside. 

If the shape of the field admits 
of it, it will be preferable to measure 
four lines about the field in such di- 
rections as to inclose it in a rectan- 
gle, and to measure offsets from the sides of this to the angles of 
the field. 



Fig. S3. 










OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 

105. In the practice of the various methods of surveying 
which have been explained, the hills and valleys which are to be 
crossed, the sheets of water which are to be passed over, the woods 
and houses which are to be gone through — all these form obstacles 
to the measurement of the necessary lines which are to join certain 
points, or to be prolonged in the same direction. Many special 
precautions and contrivances are therefore rendered necessary ; 
and the best methods to be employed, when the chain alone is to 
be used, will now be given. 

These methods for overcoming the various obstacles met with 
in practice constitute a Laxd-Geometry. Its problems are per- 



OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 77 

formed on the ground instead of on paper ; its compasses are a 
chain fixed at one end and free to swing around with the other ; 
its scale is the chain itself ; and its ruler is the same chain 
stretched tight. Its advantages are that its single instrument 
(or a substitute for it, such as a tape, a rope, etc.) can be 
found anywhere ; and its only auxiliaries are equally easy to ob- 
tain, being a few straight and slender rods, and a plumb-line, 
for which a pebble suspended by a thread is a sufficient substi- 
tute. * 

Many of these problems require the employment of perpendicu- 
lar and parallel lines. For this reason we will commence with this 
class of problems. 

The demonstrations of most of these problems will be left as 
exercises for the student. 

The elegant " Theory of Transversals " (Appendix B) will be 
an important element in some of these demonstrations. 

Problems on Perpendiculars* 
Problem 1. To erect a perpendicular at any point of a line. 

106. First Method. Let A be the point at which a perpen- 
dicular to the line is to be set out. Meas- 
ure off equal distances A B, AC, on each 
side of the point. Take a portion of the 
chain not quite 1-j- time as long as A B or 
A C, fix one end of this at B, and describe 
an arc with the other end. Do the same 
from 0. The intersection of these arcs will 
fix a point D. AD will be the perpendicu- 
lar required. Eepeat the operation on the other side of the line. 

* Many of these methods would seldom be required in practice, but cases some- 
times occur, as every surveyor of much experience in field-work has found to his seri- 
ous inconvenience, in which some peculiarity of the local circumstances forbids any 
of the usual methods being applied. In such cases the collection here given will be 
found of great value. 
, In all the figures, the given and measured lines are drawn with fine full lines, the 
visual lines, or lines of sight, with broken lines, and the lines of the result with heavy 
full lines. The points which are centers around which the chain is swung are in- 
closed in circles. The alphabetical order of the letters attached to the points shows 
in what order they are taken. 




78 



LANDSTTR VEYING. 



If that is impossible, repeat it on the same side with a different 
length of chain. 

107. Second Method, Measure off as before, equal distances A B, A 
but each about only one third of the chain. Fasten 
the ends of the chain with two pins at B and 0. 
Stretch it out on one side of the line and put a pin 
at the middle of it, D. Do the same on the other 
side of the line, and set a pin at E. Then is D E a 
perpendicular to B C. If it is impossible to perform 
the operation on both sides of the line, repeat it 
on the same side with a different length of chain, 
as shown by the lines BE and CF in the figure, 

so as to get a second point. 

108. Other Methods. All the methods to be given for the next prob- 
lem may be applied to this. 

Problem 2. To erect a perpendicular to a line at a given 
point, when the point is at or near the end of the line. 




Fig. 86. 



B 




109. First Method, Measure 40 links along the line. Let one 
assistant hold one end of the chain at 
that point ; let a second hold the 20- 
link mark which is nearest the other 
end, at the given point A, and let a 
third take the 50-link mark, and 
tighten the chain, drawing equally on 
both portions of it. Then will the 
50 link mark be in the perpendicu- 
lar desired. Eepeat the operation on 
the other side of the line so as to test the work. 

The above numbers are the most easily remembered, but the 
longer the lines measured the better ; and nearly the whole chain 
may be used ; thus : Fix down the 36th link from one end at A, 
and the 4th link from the same end on the line at B. Fix the 
other end of the chain also at B. Take the 40th-link mark from 
this last end, and draw the chain tight, and this mark will be in 
the perpendicular desired. The sides of the triangle formed by 
the chain will be 24, 32, and 40. 



OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 79 



110. Otherwise : using a 50-feet 
tape, hold the 16-feet mark at A ; 
hold the 48-feet mark and the ring- 
end of the tape together on the line ; 
take the 28-feet mark of the tape, and 
draw it tight ; then will the 28-feet 
mark be in the perpendicular desired. 



Fig. 87. 




Fig. 88. 



111. Second Method. Hold one end of the chain at A and fix 
the other end at a point B, taken at will. 
Swing the chain around B as a center, till it 
again meets the line at 0. Then carry the 
same end around (the other end remaining 
at B) till it comes in the line of C B at D. 
A D is the perpendicular required. 

Problem 3. To erect a perpendicular to 
an inaccessible line, at a given point of it. 




112. First Method. Get points in the direction of the inacces- 
sible line prolonged, and from them set out a parallel to the line, 
by methods which are given in Art. 121, etc. Find by trial the 
point in which a perpendicular to this second line (and therefore 
to the first line) will pass through the required point. 

Problem 4. To let fall a perpendicular from a given point to 
a given 



113. First Method. Let P be the given point, and AB the 
given line. Measure some distance, a 
chain or less, from C to P, and then 
fix one end of the chain at P, and 
swing it around till the same distance 
meets the line at some point D. The 
middle point E of the distance CD 
will be the required point, at which 
the perpendicular from P would meet the line. 




so 



LAND-SUB VEYING. 



114. Second Method. Stretch a chain, or a portion of it, 
from the given point P, to some point, as A, 
of the given line. Hold the end of the dis- 
tance at A, and swing round the other end of 
the chain from P, so as to set off the same 
distance along the given line from A to some 
point B. Measure BP. Then will the dis- 
tance B C from B to the foot of the desired joerpendicular 
_ BP 2 . 
~2 AB 

Problem 5. To let fall a perpendicular to a line from a point 
nearly opposite to the end of the line. 




115. First Method. Stretch a chain 
from the given point P, to some point, 
as A, of the given line. Fix to the ground 
the middle point B of the chain A P, and 
swing around the end which was at P, or 
at A, till it meets the given line in a point 
C, which will be the foot of the required 
perpendicular. 



Fig. 91. 




Fig. 92. 




Fig. 93. 




D\ 



E 






116. Second Method. At any convenient 
point, as A of the given line, erect a perpen- 
dicular of any convenient length, as AB, and 
mark a point C on the given .line in the line 
of P and B. Measure C A, C B, and P. 
Then the distance from C to the foot of the 

C A X C P 



perpendicular, i. e., CD 



CB 



Problem 6. To let fall a perpendicular to 
a line from an inaccessible point. 



^T 117. First Method. Let P be the given 

point. At any point A, on the given line, 
set out a perpendicular, AB, of any conven- 
ient length. Prolong it on the other side of 



OBSTACLES TO MEASUREMENT IN CEAIN SURVEYING. 81 



the line the same distance. Mark on the given line a point D in 
the line of P B, and a point E in the line of P C. Mark the 
point F afc the intersection of D C and B E prolonged. The line 
F P is the line required, being perpen- 
dicular to the given line at the point G. 

118. Second Method. Let A and B be 
two points of the given line. From A 
let fall a perpendicular, A C, to the vis- 
ual line, B P ; and from B let fall a per- 
pendicular, B D, to the visual line, A P. 
Find the point at which these perpendicu- 
lars intersect, as at E, and the line P E, prolonged to F, will give 
the perpendicular required. 

Problem 7. To let fall a perpendicular from a given point to 
an inaccessible line. 




119. First Method. 



Fig. 95. 




Let P be the given point, and AB the 
given line. By the preceding prob- 
lem let fall perpendiculars from A to 
B P at C ; and from B to A P at D ; 
the line P E, passing from the given 
point to the intersection of these 
perpendiculars, is the desired perpen- 
dicular to the inaccessible line A B. 
This method will apply when only two points of the line are 
visible. 

The proof of 118 and 119 is found in the "Theory of Trans- 
versals," Corollary 3. 

120. Second Method. Through the given point set out a line 
parallel to the inaccessible line. At the given point erect a per- 
pendicular to the parallel line, and it will be the required perpen- 
dicular to the inaccessible line. 



Problem 1. 



Problems on Parallels. 
To run a line from a given point parallel to a given 



line. 



82 



LAND-SUB YEYING. 



121. First Method. Let fall a perpendicular from the point to 
the line. At another point of the line, as far off as possible, erect 
a perpendicular equal in length to the one just let fall. The line 
joining the end of this line to the given point will be the parallel 
required. 

122. Second Method. Measure from 
P to any point, as C of the given line, 
and put a mark at the middle point D of 
that line. From any point, as E of the 
given line, measure a line to the point D, 
and continue it till D F = D E. Then 
will the line P F be parallel to A B. 




Fig. 97. 
C 



E 



y& 



123. Third Method. From any point, as C of the line, set off 
equal distances along the line to D and 
E. Take a point F, in the line of 
PD. Stake out the lines FC and ^— ^r 
FE, and also the line EP, crossing 

the line C F in the point G. Last- r\' ; -¥h 

ly, prolong the line D G till it meets \ i / 

the line E F in the point H. P H is p 

the parallel required. 

The proof is found in Corollary 4 of "Transversals." 

Problem 2. To run a line from a given point parallel to an in- 
accessible line. 



124. First Method. 



Fig. 98. 




Let A B be the given line, and P the 
given point. Set a stake at C, in 
the line of PA, and another at any 
convenient point, D. Through P 
set out, by the preceding problem, 
a parallel to D A, and set a stake 
at the point, as E, where this par- 
allel intersects D C prolonged. 
Through E set out a parallel to B D, 

and set a stake at the point F, where this parallel intersects 

B C prolonged. P F is the parallel required. 



F w 




OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 83 

125. Second Method. Set a stake at 
any point C in the line of A P, and an- 
other at any convenient place, as at D. 
Through P set out a parallel to A D, in- 
tersecting C D in E. Through E set out 
a parallel to D B, intersecting C B in F. 
The line P F will be the parallel required. 

126. Alinement and Measurement. We are now prepared, hav- 
ing secured a variety of methods for setting out Perpendiculars and 
Parallels in every probable case, to take up the general subject of 
overcoming Obstacles to Measurement. 

Before a line can be measured its direction must be determined. 
This operation is called Ranging the line, or Alining it, or Boning 
it.* The word alinement \ will be found very convenient for ex- 
pressing the direction of a line on the ground, whether between 
two points or in their direction prolonged. 

This branch of our subject naturally divides itself into two 
parts, the first of which is preliminary to the second, viz. : 

I. Of Obstacles to Alinement ; or lioiv to establish the direction 
of a line in any situation. 

II. Of Obstacles to Measurement; or how to find the length of a 
line which can not be actually measured. 

1. Obstacles to Alinement. 

127. All the cases which can occur under this head may be 
reduced to two, viz. : 

A. To find points in a line beyond the given points, i. e., to 
prolong the line. 

B. To find points in a line between two given points of it, i. e., 
to interpolate points in the line. 

A. To Prolong a Line. 

128. By ranging with Rods. When two points in a line are 
given, and it is desired to prolong the line by ranging it out with 



* This word, like many others used in engineering, is derived from a French word, 
borner, to work out or limit, indicating that the Normans introduced the art of sur- 
veying into England. \ Slightly modified from the French alignement. 



84 LAND-SURVEYIXG. 

rods, three persons are required, each furnished with a straight, 
slender rod, and with a plumb-line, or other means of keeping 

their rods vertical. One holds 
FlG - ^ 00 : vv his rod at one of the given 

r"'"'""^C&?P%^^^TO^" points > A in the figure > and 

^T^^<^^uX^ s --' another at B. A third, C, 

goes forward as far as he can 
without losing sight of the first two rods, and then, looking back, 
puts himself "in line" with A and B — i. e., so that when his 
eye is placed at the rod at B hides or covers the rod at A. 
This he can do most accurately by holding a plumb-line before 
his eye, so that it shall cover the first two rods. The lower end of 
the plumb-bob will then indicate the point where the third rod 
should be placed, and so with the rest. The first man, at A, is 
theD signaled and comes forward, passes both the others, and puts 
himself at D, "in line" with C and B. The man at B then goes 
on to E, and " lines " himself with D and C ; and so they proceed, 
in this "hand-over-hand" operation, as far as is desired. Stakes 
are driven at each point in the line as soon as it is determined. 

129. The rods should be perfectly straight, either cylindrical 
or polygonal, and as slender as they can be without bending. 
They should be painted in alternate bands of red and white, each 
a foot or link in length. Their lower ends should be pointed with 
iron, and a projecting bolt of iron will enable them to be pressed 
down by the foot into the earth, so that they can stand alone. 
When this is done, one man can range out a line. A rod can be set 
perfectly vertical by holding a plumb-line before the eye at some 
distance from the rod, and adjusting the rod so that the plumb- 
line covers it from top to bottom, and then repeating the operation 
in a direction at right angles to the former. A stone dropped 
from top to bottom of the rods will approximately attain 
the same end. <\/^ 

When the lines to be ranged are long, and great ac- 
curacy is required, the rods may have attached to them 
plates of tin with openings cut out of them, and black 
horse-hairs stretched from top to bottom of the openings. 



OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 85 

A small telescope must then be used for ranging these hairs in 
line. In a hasty survey, straight twigs, with their tops split to 
receive a paper folded as in the figure, may be used. 



Fig. 102. 



B 

LI 

D 



§f_L_ 

F 



H 

T 

E 



130. By Perpendiculars. The straight line, A B in the figure, is 
supposed to be stopped by a tree, a house, or other obstacle, and it 
is desired to prolong the line 
beyond this obstacle. From 

any two points, as A and B of A. 

the line, set off (by some of q"" 

the methods which have been 

given) equal perpendiculars, A C and B D, long enough to pass 
the obstacle. Prolong this line beyond the obstacle, and from 
any two points in it, as E and F, measure the perpendiculars 
E G- and F H equal to the first two, but in a contrary direction. 
Then will G and H be two points in the line A B prolonged which 
can be continued by the method of the last article. The points 
A and B should be taken as far apart as possible, as should also 
the points E and F. Three or more perpendiculars on each side 
of the obstacle may be set off, in order to increase the accuracy 
of the operation. The same thing may also be done on the other 
side of the line, as another confirmation or test of the accuracy 
of the prolonged line. 



131. By Equilateral Triangles. The obstacles noticed in the last 
article may also be overcome by means of three equilateral trian- 
gles formed by the chain. Fix one end 
of the chain, and also the end of the 
first link from its other end, at B ; fix 
the end of the 33d link at A ; take 
hold of the 66th link and draw the 
chain tight, pulling equally on each 
part, and put a pin at the point thus 
found, C in the figure. An equilateral 
triangle will thus be formed, each side 
being 33 links. Prolong the line AC past the obstacle to some 




86 



LAND-SUR VEYING. 



point, as D. Make another equilateral triangle, DEF, as be- 
fore, and thus fix the point F. Prolong DP to a length equal 
to that of AD, and thus fix a point, G. At G form a third 
equilateral triangle, GHK, and thus fix a point, K. Then will 
K G give the direction of A B prolonged. 

132. By Symmetrical Triangles. 



B 



Fig. 104. 



r 5 ^ 







Let A B be the line to be pro- 
longed. Take any convenient 
point, as C. Kange out the 
line, AC, to a point A', such 
thatCA' = CA. Eange out 
C B, so that C B' = C B. 
Eange backward A' B' to some 
point D, such that D pro- 
longed will pass the obstacle. 
Find, by ranging, the inter- 
section at E of D B and A C. From C measure, on C A', the dis- 
tance E' = C E. Then range out D C and B' E' to their intersec- 
tion in P, which will be a required point in the direction of A B 
prolonged. The symmetrical points are marked by corresponding 
letters. Several other points should be obtained in the same 
manner. 

In this, as in all similar operations, very acute intersections 
should be avoided as far as possible. 

133. By Transversals. Let AB be 

the given line. Take any two points 
and D, such that the line C D will 
pass the obstacle. Take another point, 
E, in the intersection of C A and D B. 
Measure A E, A C, C D, B D, and B E. 
Then the distance from D to P, a 
point in the required prolongation, will 

CDxBDxAE 
DeU BEXAC-BDXAE* 

Other points in the prolongation 
may be obtained in the same manner, 
by merely moving the single point C in 



Fig. 105. 




OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 87 

the line of E A ; in which case the new distances, C A and C D, 

will alone require to be measured. 

P T) v B T) 
If AE be made equal to AC, then is D P = ^ -^ __^ -^ 

If B E be made equal to B D, then is D P = a n __ r|- 

The minus sign in the denominators must be understood as 
only meaning that the difference of the two terms is to be taken, 
without regard to which is the greater. 

134. By Harmonic Conjugates. Let AB be the given line. Set 
a stake at any point C. Set stakes 

at points D, on the line C A, and ' _ 

at E, on the line C B ; these points, A b5t?-\C%P 

D and E, being so chosen that the \^*-^ v / !§:??*£•'''/ 

line DE will pass beyond the ob- \ ■*%>-. U' 2 '" / 

stacle. Set a fourth stake, F, at \ /,*•*' :"; / 

the intersection of the lines AE -^ -.. \;i / 

and DB. Set a fifth stake, G, \%;k 

anywhere in the line C F ; a sixth k\\: 

stake, H, at the intersection of C B £ 
and D G prolonged ; and a seventh, 

K, at the intersection of C A and E G prolonged. Finally, range 
out the lines D E and K H, and their intersection at P will be in 
the line A B prolonged. 

135. By the Complete Quadrilateral. Let A B be the given line. 

Take any convenient point 

IG * ' ~ C ; measure from it to B, 

and onward, in the same 

line prolonged, an equal 

distance to D. Take any 

\/- "'&/' other convenient point, E, 

D H such that CE and DE 

produced will clear the ob- 
stacle. Measure from E to A, and onward, an equal distance, to 
F. Range out the lines F C and D E to their intersection in G. 



88 



LAND-SUE VEYING. 



Front Vi 



Fig. 108. 

Side View. Back Vieio. 




m 




Bange out F D and C E to intersect in H. Measure G H. Its 
middle point, P, is the required point in the line of A B prolonged. 
The unavoidable acute intersections in this construction are ob- 
jectionable. 

B. To INTERPOLATE POINTS IN A LlNE. 

136. The most distant given point of the line must be made 

as conspicuous as 
possible by any 
efficient means, 
such as placing 
there a staff bear- 
ing a flag : red and 
white, if seen 
against woods or 
other dark back- 
ground ; and red 
and green, if seen 
against the sky. 
A convenient 
and portable signal is shown in the figure. 

The figure represents a disk of tin about six inches in diameter, 
painted white and hinged in the middle, to make it more portable. 
It is kept open by the bar, B, being turned into the catch, C. A 
screw, S, holds the disk in a slit in the top of the pole. 

Another contrivance is a strip of tin, which has its ends bent 
horizontally in contrary directions. As the wind will take strong- 
est hold of the side which is concave toward it, the bent strip will 
continually revolve, and thus be very conspicuous. Its upper half 
should be painted red, and its lower half white. 

A bright tin cone set on the staff can be seen at a great distance 
when the sun is shining. 



137. Banging to a point thus made conspicuous is very simple 
when the ground is level. The surveyor places his eye at the near- 
est end of the line, or stands a little behind a rod placed on it, and 
by signs moves an assistant, holding a rod at some point as nearly 



OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 89 

in the desired line as he can guess, to the right or left, till his rod 
appears to cover the distant point. 




-~^B 



i38. Across a Valley. When a valley or low spot intervenes be 
tween the two ends of the 

line, A and Z in the figure, ^ FlG - 109 - 

a rod held in the low place, ft^i 

as at B, would seldom be 
high enough to be seen 
from A, to cover the dis- 
tant rod at Z. In such a 
case, the surveyor at A 

should hold up a plumb-line over the point, at arm's length, and 
place his eye so that the plumb-line covers the rod at Z. He 
should then direct the rod held at B to be moved till it, too, 
is covered by the plumb-line. The point B is then said to be 
"in line" between A and Z. In geometrical language, B has now 
been placed in the vertical plane determined by the vertical plumb- 
line and the point Z. Any number of intermediate points can thus 
be "interpolated," or placed in line between A and Z. 

139. Over a Hill. When a hill rises between two points and 

prevents one being seen from the other, as in the figure (the upper 

„ ,„„ part of which 

Fig. 110. 



shows the hill in 
"elevation, "and 
the lower part 
in "plan"), two 
observers, B and 
C, each holding 
a rod, may place 
themselves on 
the ridge, in the 
line between the 
two points, as 
nearly as they 
can guess, and so that each can at once see the other and the point 




90 



LAND-SUB VEYim. 



beyond him. B looks to Z, and by signals puts C "in line." 
C then looks to A, and puts B in line at B'. B repeats his oper- 
ation from B', putting C at C, and is then himself moved to B", 
and so they alternately ' ' line " each other, continually approxi- 
mating to the straight line between A and Z, till they at last find 
themselves both exactly in it, at B'" and 0"'. 

140. A single person may put himself in line between two 
points, on the same principle, by laying a straight stick on some 
support, going to each end of it in turn, and making it point suc- 
cessively to each end of the line. The " Surveyor's Cross," Art. 
93, is convenient for this purpose, when set up between the two 
given points and moved again and again, until, by repeated trials, 
one of its slits sights to the given points when looked through in 
either direction. 



Fig. 111. 



MPN 



141. On Water. A simple instrument for the same object is 
represented in the figure. A B and C D 
are two tubes, about 1£ inch in diame- 
ter, connected by a smaller tube, EF. 
A piece of looking-glass, G H, is placed 
in the lower part of the tube A B, and 
another, KL, in the tube CD. The 
planes of the two mirrors are at right 
angles to each other. The eye is placed 
at A, and the tube A B is directed to 
any distant object, as X, and any other 
object behind the observer, as Z, will be 
seen, apparently under the first object 
a in the mirror G H, by reflection from 

the mirror K L, when the observer has 
succeeded in getting in line between the two objects. M X are 
screws by which the mirror K L may be adjusted. The distance 
between the two tubes will cause a small parallax, which will, 
however, be insensible except when the two objects are near to- 
gether. 





OBSTACLES TO MEASUREMENT IN CHAm-SURVEYWG. 91 

142. Through a Wood. When a wood intervenes between any 

two giyen points, preventing one from being seen from the other, 

as in the figure, in which A and Z are the given points, proceed 

thus : Hold a rod 

, . , Fig. 112. 

at some point K 

B' as nearly in ^ _ d 

the desired line JLs^Ss 

from A as can 
be guessed at, 

and as far from A as possible. To approximate to the proper 
direction, an assistant may be sent to the other end of the line, 
and his shouts will indicate the direction ; or a gun may be 
fired there ; or, if very distant, a rocket may be sent up after 
dark. Then range out the " random line" AB', by the method 
given in Art. 128, noting also the distance from A to each point 
found, till you arrive at a point Z', opposite to the point Z — i. e., 
at that point of the line from which a perpendicular there erected 
would strike the point Z. Measure Z' Z. Then move each of the 
stakes, perpendicularly from the line A Z', a distance proportional 
to their distances from A. Thus, if AZ' be 1,000 links, and Z'Z 
be 10 links, then a stake B', 200 links from A, should be moved 2 
links to a point B, which will be in the desired straight line A Z ; 
if C be 400 links from A, it should be moved 4 links to C, and 
so with the rest. The line should then be cleared, and the accu- 
racy of the position of these stakes tested by ranging from A to Z. 



143. To an Invisible Intersection. Let AB and CD be two 

lines, which, if pro- 

IG. 113. 



y longed, would meet m a 

/\ ^^W^^^^Ca:^^^' point Z, invisible from 

\ \y \ if^fe^lS^^ ^ either of them ; and let 

P be a point from which 
a line is required to be 
set out tending to this 
invisible intersection. 
Set stakes at the five 
given points, A, B, C, D, P. Set a sixth stake at E, in the 
7 




9 2 LANB-SUB V EYING. 

alinements of A D and C P ; and a seventh stake at F, in the 
alinements of B C and A P. Then set an eighth stake at G-, in the 
alinements of B E and DF. P G- will be the required line. This 
is an application of the " Theory of Transversals." 

Otherwise: Through P range out a parallel to the line BD. 
Note the points where this parallel meets A B and C D, and call 
these points Q and E. Then the distance from B, on the line B D, 
to a point which shall be in the required line running from P to 

«. • • -w • + -ii i, BDxQP 
the invisible point, will be = ^5 . 

II. Obstacles to Measurement. 

144. The cases in which the direct measurement of a line is 
prevented by various obstacles may be reduced to three : 

A. When loth ends of the line are accessible. 

B. When one end of it is inaccessible. 

C. When both ends of it are inaccessible. 

A. When Both Ends of the Line aee accessible. 

145. By Perpendiculars. On reaching the obstacle, as at A in 

the figure, set off a perpendicular, 
A B ; turn a second right angle at 
B, and measure past the obstacle ; 
turn a third right angle at C, and 
measure to the original line at D. 

Then will the measured distance, 
B C, be equal to the desired distance, A D. 

If the direction of the line is also Fig. 115. 

unknown, it will be most easily obtained -^ 

by the additional perpendiculars shown — fB — 

in Fiff. 102 of Art. 130. \ / Ar\ / 



Fig. 114. 




146. By Equilateral Triangles. The C \ 

method given in Art. 131 for determin- 
ing the direction of a line through an 
obstacle will also give its length ; for 
in Fig. 115 the desired distance A G is 
equal to the measured distances AD or D G. 



11 




F 



OBSTACLES TO MEASUREMENT IN CHAIN-SURYEYING. 93 



Let A B be the distance re- 



Fig. 116. 




147. By Symmetrical Triangles, 
quired. Measure from A obliquely 
to some point past the obstacle. 
Measure onward, in the same line, 
till C D is as long as A C. Place 
stakes at and D. From B meas- 
ure to C, and from measure on- 
ward, in the same line, till C E is 
equal to OB. Measure E D, and 
it will be equal to A B, the distance 

required. If more convenient, make CD and CE equal, respec- 
tively, to half of A and C B ; then will A B be equal to twice 
DE. 

Let A B be the required distance. Set 
a stake, C, in the line prolonged ; 
set another stake, D, so that C 
and B can be seen from it ; and a 
third stake, E, in the line of B D 
prolonged, and at a distance from 
D equal to the distance from D 
to B. Set a fourth stake, F, at 
the intersection of EA and CD. 




Measure AC, A F, and F E. 



Then is AB = ^(FE-AF). 



B. When One End of the Line is inaccessible. 
149. By Perpendiculars. This principle may be applied in a 
variety of ways. In Fig. 118 let AB be the 
required distance. At the point A set off AC 
perpendicular to AB, and of any convenient 
length. At C set off a perpendicular to C B. 
and continue it to a point, D, in the line of A 

AC 2 



Fig. 118. 

A 



and B. Measure D A. Then is A B 



AD. 




150. Otherivise : At the point A, in Fig. 119, 
set off a perpendicular, AC. At C set off another perpendicu- 



94 LAXD-SUR VETING. 

lar, CD. Find a point, E, in the line of 
AC and B D. Measure AE and E C. Then 
AEXCD 



isAB = 




/ 



CE ' 

If E C be made equal to A E, and D be 
set in the line of B E, and also in the per- 
pendicular from C, then will C D be equal 
to AB. 



If EC = iAE, then CD = *AB. 

151. Otherwise : At A, in Fig. 120, measure 
a perpendicular, A C, to the line AB ; and at 
any point, as D in this line, set off a perpen- 
dicular to D B, and continue it to a point E, in 
the line of C B. Measure DE and also DA. 

. ■ ACXAD 
Then is ABs^^TaC' 





152. By Parallels. From A measure 

AC in any convenient direction. From a 

point D, in the line of B C, measure a 

line parallel to C A, to a point E in the 

line of A B. Measure also A E. 

. . _ ACXAE 
Then is AB = ^ 1 - : — ^. 



153. By a Parallelogram. Set a stake, C, in the line of A and 
B, and set another stake, D, wherever conven- 
ient. AYith a distance equal to CD, describe 
from A an arc on the ground ; and, with a dis- 
tance equal to A C, describe another arc from 
D intersecting the first arc in E. Or, take A C 
and CD so that together they make one chain ; 
fix the ends of the chain at A and D ; take 
hold of the chain at such a link that one part 
of it equals A C and the other CD, and draw 
it tight to fix the point E. Set a stake at F in the intersection 




OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 95 



of A E and D B. 
AC X AF 



EF 



; orCB = 



Measure A F and E F. 
ACxCD 



Then is A B = 



EF 



Fig. 123. 



154. By Symmetrical Triangles. Let AB be the required dis- 
tance. From A measure a line in any- 
convenient direction, as A C, and meas- 
ure onward, in the same direction, till 
CD=AO. Take any point E in the 
line of A and B. Measure from E to 
C, and onward in the same line, till C F 
= C E. Then find by trial a point G, 
which shall be at the same time in the 
line of C and B, and in the line of D 
and F. Measure the distance from G 
to D, and it will be equal to the re- 
quired distance from A to B. If more convenient, make CD = 
| A 0, and C F = J E, as shown by the finely dotted lines in 
the figure. Then will D G = J A B. 




CE 
CA 



Fig. 124. 



155. Othenvise : Prolong B A to some point C. Eange out 
any convenient line C A', and measure 
C A ' = C A. The triangle C A' B is 
now to be reproduced in a symmetrical 
triangle situated on the accessible 
ground. For this object take, on A C, 
some point D and measure CD' = 
C D. Find the point E at the inter- 
section of AD' and A' D. Find the 
point F at the intersection of A' B and 
Lastly, find the point B' at the intersection of A F and 
Then will A' B' = A B. The symmetrical points have cor- 




responding letters affixed to them. 



156. By Transversals. Set a stake, C, in the alinement of 
B A ; a second, D, at any convenient point ; a third, E, in 
the line D ; and a fourth, F, at the intersection of the aline- 



96 



LAND-SUE VEYIXa. 




D^ 



Fig. 125. ments of D A and E B. Measure A C, 

CE,ED,DF, and F A. Then is A B = 

AC X AF XDE 
CExDF-AFxDE* 

If the point E be taken in the middle 
of C D (as it is in the figure), then A B = 
ACX AF 
DF-AF' 

If the point F be taken in the middle of A D, then A B = 
AC X DE 
CE-DE' 

The minus signs must be interpreted as in Art. 121. 

157. By Harmonic Division. Set stakes, C and D, on each side 
of A, and so that the three are in the 
same straight line. Set a third stake 
at any point, E, of the line A B. Set 
a fourth, F, at the intersection of C B 
and D E ; and a fifth, G, at the inter- 
section of DB and C E. Set a sixth 
stake, H, at the intersection of A B and 
F G. Measure A E and E H. Then is 

AE X AH ' 



Fig. 126. 



B 



AB = 



AE-EH' 




Fig. 127. 



158. To an Inaccessible Line. The shortest distance, C D, from 
a given point, C, to an inaccessible 
straight line A B, is required. From 
C let fall a perpendicular to A B, by 
the method of Art 119. Then set a 
stake at any point, E, on the line A C ; 
set a second, F, at the intersection of 
E B and C D ; a third, C, at the inter- 
section of AF and CB; and a fourth, 
H, at the intersection of E G and C D. 
CH x CF 




Measure C H and H F. 
CH + HF 



Then is C D = 



CH-HF 



or CD 



= CH. 



CH-HF 



or CD = 



CHX CF 

2CH-CF 



OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 97 



When two lines (as A B, 



Fig. 128. 




159. To an Inaccessible Intersection. 
C D, in the figure) meet in a 
river, a building, or any other 
inaccessible point, the distance 
from any point of either to 
their intersection, D E, for ex- 
ample, may be found thus : 
From any point B, on one line, 
measure B D, and continue it 
till DF = DB. From any 

other point G of the former line measure G D, and continue the 
line till D H = G D. Continue H F to meet D in some point 
K. Measure K D. KD will be equal to the desired distance D E. 

B E can be found by measuring F K, which is equal to it. 

If D F and D H be made respectively equal to one half or one 
third, etc., of D B and D G, then will K D and K F be respectively 
equal to one half or one third, etc., of D E and BE. 

0. When Both Ends of the Line aee inaccessible. 

160. By Similar Triangles. Let A B be the inaccessible dis- 

tance. Set a stake at any convenient point 
Fro. 129. C, and find the distances C A and C B by 

any of the methods just given. Set a sec- 
ond stake at any point, D, on the line C A. 

CB X CD 




Measure a distance equal to 



CA 



from C, on the line C B, to some point E. 



Measure D E. Then is A B = 



AC XDE 
CD ' 



If more convenient, measure C D in the contrary direction from 
the river, as in Fig. 130, instead of toward it, and in other respects 
proceed as before. 



161. By Parallels. Let A B be the in- 
accessible distance. From any point, as C, 
range out a parallel to A B, as in Art. 
124, etc. Find the distance C A by Art. 
149, etc. Set a stake at the point E, the 



Fro. 130. 




98 



LAND-SUE VEYING. 




intersection of C A and D B, and measure 
CE. 



_ . . B CD X ( A - C E) 
Then is A B = hr^ 

Kj Hi 



'Fig. 132. 



162. By a Parallelogram. Set a stake at 
any convenient point C. Set stakes D and 
E anywhere in the alinements C A and C B. 

With D as a center, and a length 

of the chain equal to C E, describe 

an arc ; and with E as a center, and 

a length of the chain equal to C D, 

describe another arc, intersecting the 

former one at F. A parallelogram, 

C D E F, will thus be formed. Set 

stakes at G and H, where the aline- 
ments D B and E A intersect the sides of this parallelogram. 

Measure CD, D F, G F, F H, and H G. The inaccessible dis- 




tance A B = 



CD X DFX GH 
FGxFH * 



If C D = C E, then A B = 



CD'XGH 
FGxFH* 



163. By Symmetrical Triangles. 



Fig. 133. 



Take any convenient point, as 
C. Set stakes at two other 
points, D and D', in the same 
line, and at equal distances 
from C. Take a point E, in 
the line of AD; measure 
from it to C, and onward till 
CE' = CE. Take a point 
F in the line of BD; meas- 
ure from it to C, and onward 
till CF = CF. Bange out 
the lines A C and E' D', and 
set a stake at their intersec- 
tion, A'. Bange out the 

lines B C and F' D'; and set a stake at their intersection, B'. 

Measure A' B'. It will be equal to the desired distance A B. 




OBSTACLES TO MEASUREMENT IN CHAIN-SURVEYING. 99 



Fig. 134. 



. 164. Otherivise : Take any convenient point, as C, and set off 
equal distances on each side 
of it, in the line of C A, to 
D and D'. Set off the same 
distances from C, in the 
line of C B, to E and E'. 
Through set out a parallel 
to DE or D'E', and set 
stakes at the points F and 
F' where this parallel in- 
tersects A E' and BD'. 
Kange out the lines A D' 
and E F', and set a stake 
at their intersection A'. 

Eange out the lines B E' and D F ? and set a stake at their inter- 
section B'. Measure A' B', and it will be equal to the desired 
distance A B. 




CHAPTER III. 

COMPASS- SURVEYING J OR BY THE THIRD METHOD. 

165. Angular Surveying determines the relative positions of 
points, and therefore of lines, on the Third Principle, as ex- 
plained in Art. 5. 

Either the compass or the transit may be employed in angular 
surveying. 

166. Surveying with the compass is a less direct operation 
than surveying with the transit. But as the use of the com- 
pass is much more rapid and easy, for this reason, as well as 
for its smaller cost, it is the instrument most commonly em- 
ployed in land-surveying in spite of its imperfections and in- 
accuracies. 

The method of Polar Surveying (or surveying by the third 
method) embraces two minor methods. The most usual one 
consists in going around the field with the instrument, setting it 
at each corner, and measuring there the angle which each side 
makes with its neighbor, as well as the length of each side. This 
method is called by the French the method of Cheminement. It 
has no special name in English, but may be called (from the 
American verb, to progress) the Method of Progression. The 
other system, the Method of Radiation, consists in setting the in- 
strument at one point and thence measuring the direction and 
distance of each corner of the field or other object. The corre- 
sponding name of what we have called triangular surveying is the 
Method of Intersections, since it determines points by the intersec- 
tions of straight lines. 



THE COMPASS. 



101 



167. "When the two lines which form an angle lie in the same 
horizontal or level plain, the angle is called a horizontal angle* 
When these lines lie in a plane perpendicular to the former, the 
angle is called a vertical angle. 

When one of the lines is horizontal, and the other line from tho 
eye of the observer passes above the former, and in the same verti- 
cal plane, the angle is called an angle of elevation. 

When the latter line passes below the horizontal line, 
and in the same vertical plane, the angle is called an angle 
of depression. 

When the two lines which form an angle lie in other 
planes which make oblique angles with each of the former 
planes, the angle is called an oblique angle. 

Horizontal angles are the only angles employed in com- 
mon land-surveying. 

Fig. 135. 




THE COMPASS. 

168. The Needle. The most essential part of the compass is 
the magnetic needle. It is a slender bar of steel, usually five or 
six inches long, strongly magnetized, and balanced on a pivot, so 
that it may turn freely, and thus be enabled to continue pointing 
in the same direction (that of the "magnetic meridian" approxi- 
mately north and south) however much the "compass-box," to 
which the pivot is attached, may be turned around. 

As it is important that the needle should move with the least 

* A plane is said to be horizontal or level when it is parallel to the surface of 
standing water, or perpendicular to a plumb-line. A line is horizontal when it lies 
in a horizontal plane. 



102 LAND-SURVEYING. 

possible friction, the pivot should be of the hardest steel ground to 
a very sharp point ; and in the center of the needle, which is to 
rest on the pivot, should be inserted a cap of agate, or other hard 
material. Iridium for the pivot, and ruby for the cap, are still 
better. 

If the needle be balanced on its pivot before being magnetized, 
one end will sink, or " dip," after the needle is magnetized. To 
bring it to a level, several coils of wire are wound around the 
needle so that they can be slid along it, to adjust the weight of its 
two ends and balance it more perfectly. 

The north end of the needle is usually cut into a more orna- 
mental form than the south end for the sake of distinction. 

The principal requisites of a compass-needle are intensity of 
directive force and susceptibility. Beyond a certain limit, say five 
inches, no additional power is gained by increasing the length of 
the needle. On the contrary, longer ones are apt to have their 
strength diminished by several consecutive poles being formed. 
Short needles, made very hard, are therefore to be preferred. 

The needle should not come to rest very quickly. If it does, it 
indicates either that it is weakly magnetized, or that the friction 
on the pivot is great. Its sensitiveness is indicated by the num- 
ber of vibrations which it makes in a small space before coming 
to rest. 

A screw, with a milled head, on the under side of the plate 
which supports the pivot, is used to raise the needle off this pivot 
when the instrument is carried about, to prevent the point being 
dulled by unnecessary friction. 

169. The Sights. Next after the needle, which gives the direc- 
tion of the fixed line whose angles with the lines to be surveyed are 
to be measured, should be noticed the sights, which show the direc- 
tions of these last lines. At each end of a line passing through the 
pivot is placed a " sight," consisting of an upright bar of brass, 
with openings in it of various forms — usually slits, with a circular 
aperture at their top and bottom ; all these arrangements being 
intended to enable the line of sight to be directed to any desired 
object with precision. 



TEE COMPASS. 103 

A telescope which can move up and down in a vertical plane, 
i. e., a plunging telescope, or one which can turn completely over, 
is sometimes substituted for the sights. It has the great advantage 
of giving more distinct vision at long distances, and of admitting 
of sights up and down very steep slopes. Its accuracy of vision is, 
however, rendered nugatory by the want of precision in the read- 
ings of the needle. If a telescope be applied to the compass, a 
graduated circle with vernier should be added, thus converting the 
compass into a "transit." 

170. The Divided Circle. We now have the means of indicating 
the directions of the two lines whose angle is to be measured. The 
number of degrees contained in it is to be read from a circle divided 
into degrees, in the center of which is fixed the pivot bearing the 
needle. The graduations are usually made to half a degree, and a 
quarter of a degree or less can then be "estimated." The pivot 
and needle are sunk in a circular box, so that its top may be on a 
level with the needle. The graduations are usually made on the 
top of the surrounding rim of the box, but should also be con- 
tinued down its inside circumference so that it may be easier to 
see with what division the ends of the needle coincide. 

The degrees are not numbered consecutively from 0° around to 
360°, but ran from 0° to 90°, both ways from the two diametrically 
opposite points at which a line, passing through the slits in the 
middle of the sights, would meet the divided circle. 

The lettering of the surveyor's compass has one important dif- 
ference from that of the mariner's compass. 

When we stand facing the north, the east is on our right hand, 
and the west on our left. The graduated card of the mariner's 
compass, which is fastened to the needle and turns with it, is 
marked accordingly. But, in the surveyor's compass, one of the 
points being marked N. or north (or indicated by a fleur-de-lis), 
and the opposite one S. or south, the 90-degrees-point on the right 
of this line, as you stand at the S. end and look toward the K, is 
marked W. or west ; and the left hand 90-degrees-point is marked 
E. or east. The reason of this will be seen when the method of 
using the compass comes to be explained. 



104 



LAND-SUR YEYING. 



171. The Points. In ordinary land-surveying only fonr points 

of the compass have names, viz., north, south, east, and west ; 

the direction of a line being de- 
Fig. 136. .,',,, 

scribed by the angle which it 

makes with a north and south 
line to its east or to its west. 
But, for nautical purposes, the 
circle of the compass is divided 
into thirty-two points, the names 
of which are shown in the figure. 
Two rules embrace all the cases : 
1. When the letters indicating 
two points are joined together, 
the point half-way between the 
two meant ; thus, 2s". E. is 
half-way between north and east ; and X. X. E. is half-way be- 
tween north and northeast. 2. When the letters of two points 
are joined together with the intermediate word by, it indicates 
the point which comes next after the first in going toward the 
second ; thus, N. by E. is the point which follows north in going 
toward the east ; S. E. by S. is the next point from southeast 
going toward the south. 




172. Eccentricity. The center-pin, or pivot of the needle, 
ought to be exactly in the center of the graduated circle ; the 
needle ought to be straight, and the line of the sights ought to 
pass exactly through this center and through the points of the 
circle. If this is not the case, there will be an error in every ob- 
servation. This is called the error of eccentricity. 

When the maker of a compass is about to fix the pivot in place, 
he is in doubt of two things : whether the needle is perfectly 
straight, and whether the pivot is exactly in the center. In Figs. 
137 and 138 both of these are represented as being excessively in 
error. 

First, to examine if the needle be straight. Fix the pivot 
temporarily so that the ends of the needle may cut opposite de- 
grees — i. e., degrees differing by 180°. The condition of things at 



THE COMPASS. 



105 



this stage of progress will be represented by Fig. 137. Then turn 
the compass-box half-way around. The error will now be doubled, 



Fig. 137. 




as is shown by Fig. 138, in which the former position of the needle 
is indicated by a dotted line.* Now bend the needle, as in Fig. 
139, till it cuts divisions midway betwen those cut by it in its 
present and in its former position. This makes it certain that the 
needle is straight, or that its two ends and its center lie in the same 
straight line. 

Second, to put the pivot in the center. Move it till the 
straightened needle cuts opposite divisions. It is then certain that 
the direction of the needle passes through the center. Turn the 
compass-box one quarter around, and, if the needle does not then 
cut opposite divisions, move the pivot till it does. Repeat the 
operation in various positions of the box. It will be a sufficient 
test if it cuts the opposite divisions of 0°, 45°, and 90°. 

To fix the sights precisely in line, draw a hair through their 
slits and move them till the hair passes over the points on the 
circle. 

The surveyor can also examine for himself, by the principle of 
reversion, whether the line of the sights passes through the center 
or not. Sight to any very near object. Read off the number of 
degrees indicated by one end of the needle. Then turn the com- 
pass half around, and sight to the same object. If the two read- 
ings do not agree, there is an error of eccentricity, and the arith- 
metical mean, or half sum of the two readings, is the correct one. 

In Fig. 140 the line of sight AB is represented as passing to 

* This is another example of the fruitful principle of reversion. 



106 



LAND-SUR YEYIXG. 



one side of the center, and the needle as pointing to 46°. In Fig. 
141 the compass is supposed to have been turned half around, and 



Fig. 140. 




Fig. 141. 




the other end of the sights to be directed to the same object. 
Suppose that the needle would have pointed to 45° if the line of 
sight had passed through the center ; the needle will now point 
to 44°, the error being doubled by the reversion, and the true 
reading being the mean. 

This does not, however, make it certain that the line of the 
sights passes through the points, which can only be tested by the 
hair, as mentioned above. 



173. Levels. On the compass-plate are two small spirit-levels. 
They consist of glass tubes slightly curved upward, and nearly 
filled with alcohol, leaving a bubble of air within them. They 
are so adjusted that, when the bubbles are in the centers of the 
tubes, the plate of the compass shall be level. One of them lies 
in the direction of the sights, and the other at right angles to this 
direction. 

On the compass-plate, and between the vernier and the left- 
hand sight in the figure, is the Oufkeeper, for keeping tally of the 
chains in any distance. 

174. Tangent Scale. This is a convenient, though not essen- 
tial, addition to the compass, for the purpose of measuring the 
slopes of ground, so that the proper allowance in chaining may be 
made. In the figure of the compass may be seen, on the edge of 



THE COMPASS, 



107 



the left-hand sight, a small projection of brass with a hole through 
it. On the edge of the other sight are engrayed lines numbered 
from 0° to 20°, the 0° being of the same height above the compass- 
plate that the eye-hole is. To use this, set the compass at the bot- 
tom of a slope, and at the top set a signal of exactly the height of 
the eye-hole from the ground. Level the compass very carefully, 
particularly by the level which lies lengthwise, and, with the eye 
at the eye-hole, look to the signal and note the number of the 
division on the farther sight which is cut by the visual ray. That 
will be the angle of the slope ; the distances of the engraved lines 
from the 0° line being tangents (for the radius equal to the dis- 
tance between the sights) of the angles corresponding to the num- 
bers of the lines. 

175. Vernier. The compass-box is connected with the plate 
which carries it and the sights, so that it can turn around on this 
plate. This motion is given to it by a slow motion or tangent 
screw, shown on the left of the compass-box in the figure. The 
space through which the compass-box is moved is indicated by a 
vernier. For description of a vernier, and method of reading it, 
see subject Verniers under Transit-Surveying. 



176. Tripod. The compass, like 
most surveying instruments, is usually 
supported on a tripod, consisting of 
three legs, shod with iron, and so con- 
nected at top as to be movable in any 
direction. There are many forms of 
these. Lightness and stiffness are the 
qualities desired. The most usual form 
is shown in the figures of the transit 
and the level. Of the two represented 
in Figs. 142 and 143 the first has the 
advantage of being very easily and 
cheaply made ; and the second that of 
being light and yet capable of very 
firmly resisting horizontal torsion. 
8 



Pig. 142. 



Fig. 143. 




108 



LAND-SUR Y EYING. 



The joints by which the instrument is connected with the 
tripod are also various. Fig. 144 is the " ball-and-socket joint," 
most usual in this country. It takes its name from the ball in 
which terminates the covered spindle which enters a corresponding 
cavity under the compass-plate and the socket in which this ball 
turns. It admits of motion in any direction, and can be tightened 
or loosened by turning the upper half of the hollow piece inclosing 



Fig. 144. 




Fig. 145. 



Fig. 146. 




it, which is screwed on the lower half. Fig. 145 is called the 
"shell-joint." In it the two shell-shaped pieces inclosing the ball 
are tightened by a thumb-screw. Fig. 146 is " Cugnot's joint." 
It consists of two cylinders placed at right angles to each other, 
and through the axes of which pass bolts, which turn freely in the 
cylinder, and can be tightened or loosened by thumb-screws at 
their ends. The combination of the two motions which this joint 
permits enables the instrument which it carries to be placed in any 
desired direction. This joint is much the most stable of the 
three. 



177. Jacob's Staff. A single leg, called a "'Jacob's staff," has 
some advantages, as it is lighter to carry in the field, and can be 
made of any wood on the spot where it is to be used, thus saving 
the expense of a tripod and the trouble of its transportation. Its 
upper end is fitted into the lower end of a brass head which has a 
ball-and-socket joint and axis above. Its lower end should be shod 



THE COMPASS. 



109 



with iron, and a spike running through it is useful for pressing it 
into the ground with the foot. Of course, it can not be conven- 
iently used on frozen ground or on pavements. It may, however, 
be set before or behind the spot at which the angle is to be meas- 
ured, provided that it is placed very precisely in the line of direc- 
tion from that station to the one to which a sight is to be taken. 



Fig. 147. 



178. The Prismatic Compass. The peculiarity of this instru- 
ment (often called Schmalcalder's) is that a glass triangular prism 
is substituted for one of the sights. Such a prism has this peculiar 
property that at the same time it can be seen through, so that a 
sight can be taken through it, and that its upper surface reflects 
like a mirror, so that the numbers of the degrees immediately 
under it can be read off at the same time that a sight to any object 
is taken. Another peculiarity necessary for profiting by the last 
one is that the divided 
circle is not fixed, but is 
a card fastened to the nee- 
dle and moving around 
with it, as in the mari- 
ner's compass. The mi- 
nute description which fol- 
lows is condensed from 
Simms. 

In the figure, A repre- 
sents the compass-box and 
B the card, which, being 
attached to the magnetic 
needle, moves as it moves 
around the agate center a, on which it is suspended. The circum- 
ference of the card is usually divided to } or J of a degree. C is 
a prism which the observer looks through. The perpendicular 
thread of the sight-vane, E, and the divisions on the card appear 
together on looking through the prism, and the division with which 
the thread coincides when the needle is at rest, is the " bearing '* 
of whatever object the thread may bisect — i. e., is the angle which 
the line of sight makes with the direction of the needle. The 




110 



LAND-SUE VEYING. 



prism is mounted with a hinge-joint, D. The sight- vane has a 
fine thread stretched along its opening in the direction of its 
length, which is brought to bisect any object by turning the box 
around horizontally. F is a mirror made to slide on or off the 
sight-vane, E ; and it may be reversed at pleasure — that is, turned 
face downward ; it can also be inclined at any angle by means of 
its joint, d ; and it will remain stationary on any part of the vane 
by the friction of its slides. Its use is to reflect the image of an 
object to the eye of an observer when the object is much above or 
below the horizontal plane. The colored glasses represented at Gl- 
are intended for observing the sun. At e is shown a spring, which, 
being pressed by the finger at the time of observation and then re- 
leased, checks the vibrations of the card, and brings it more speed- 
ily to rest. A stop is likewise fixed to the other side of the box, 
by which the needle may be thrown off its center. 

The method of using this instrument is very simple : First 
raise the prism in its socket, &, until you obtain a distinct view of 
the divisions on the card. Then, standing over the point where 
the angles are to be taken, hold the instrument to the eye, and, 
looking through the slit, C, turn around till the thread in the 
sight-vane bisects one of the objects whose bearing is required ; 
then, by touching the spring, e, bring the needle to rest, and the 
division on the card which coincides with the thread on the vane 
will be the bearing of the object from the north or south points of 
the magnetic meridian. Then turn to any other object and repeat 
the operation ; the difference between the bearing of this object 

and that of the former will be the 
angular distance of the objects in 
question. Thus, suppose the 
former bearing to be 40° 30', and 
the latter 10° 15', both east or 
both west, from the north or 
south, the angle will be 30° 15'. 
The divisions are generally num- 
bered 5°, 10°, 15°, etc., around the 
circle to 360°. 

The figures on the compass- 



Fig. 148. 




THE COMPASS. HI 

card are reversed or written upside down, as in the figure (in 
which only every fifteenth degree is marked), because they arc 
again reversed by the prism. 

The prismatic compass is generally held in the hand, the bear- 
ing being caught, as it were, in passing; but more Accurate read- 
ings would, of course, be obtained if it rested on a support, such as 
a stake cut flat on its top. 

In the former mode, the needle never comes completely to rest, 
particularly in the wind. In such cases, observe the extreme divis- 
ions between which the needle vibrates, and take their arithmeti- 
cal mean. 

179. Defects of the Compass. The compass is deficient in both 
precision and correctness.* 

The former defect arises from the indefiniteness of its mode of 
indicating the part of the circle to which it points. The point of 
the needle has considerable thickness; it can not quite touch the 
divided circle ; and these divisions are made only to whole or half 
degrees, though a fraction of a division may be estimated or 
guessed at. The vernier does not much better this, as we shall see 
when explaining its use. Now, an inaccuracy of one quarter of a 
degree in an angle — i. e., in the difference of the directions of two 
lines — causes them to separate from each other 5J inches at the end 
of 100 feet ; at the end of 1,000 feet, nearly 4|- feet ; and, at the 
end of a mile, 23 feet. A difference of only one tenth of a degree, 
or six minutes, would produce a difference of If foot at the end 
of 1,000 feet ; and 9} feet at the distance of a mile. Such are 
the differences which may result from the want of precision in 
the indications of the compass. 

But a more serious defect is the want of correctness in the com- 
pass. Its not pointing- exactly to the true north does not, indeed, 
affect the correctness of the angles measured by it. But it does 
not point in the same or in a parallel direction during even the 

* The student must not confound these two qualities. To say that the sun ap- 
pears to rise in the eastern quarter of the heavens and to set in the western is correct, 
but not precise. A watch with a second-hand indicates the time of day precisely, but 
not always correctly. The statement that two and two make five is precise, but is not 
usually regarded as correct. 



112 LAND-SURVEYING. 

same day, but changes its direction between sunrise and noon 
nearly a quarter of a degree, as will be fully explained hereafter. 
The effect of such a difference we have just seen. This direc- 
tion may also be greatly altered in a moment, without the 
knowledge of the surveyor, by a piece of iron being brought near 
to the compass, or by some ofcher local attraction, as will be noticed 
in Art. 186. This is the weak point in the compass. 

Notwithstanding these defects, the compass is a very valuable 
instrument, from its simplicity, rapidity, and convenience in use ; 
and, though never precise, and seldom correct, it is generally not 
very wrong. 

THE FIELD-WORK. 

180. Taking Bearings. The "bearing" of a line is the angle 
which it makes with the direction of the needle. The bearing and 
length of a line are named collectively the Course. 

To take the bearing of any line, set the compass exactly over 
any point of it by a plumb-line suspended, from beneath the center 
of the compass, or, approximately, by dropping a stone. Level 
the compass by bringing the air-bubbles to the middle of the level 
tubes. Direct the sights to a rod held truly vertical or "plumb " 
at another point of the line, the more distant the better. The two 
ends are usually taken. Sight to the lowest visible point of the 
rod. When the needle comes to rest, note what division on the 
circle it points to ; taking the one indicated by the north end of 
the needle, if the north point on the circle is farthest from you, 
and vice versa. 

In reading the division to which one end of the needle points, 
the eye should be placed over the other end, to avoid the error 
which might result from the "parallax," or apparent change of 
place of the end read from, when looked at obliquely. 

The bearing is read and recorded by noting between what let- 
ters the end of the needle comes, and to what number ; naming, 
or writing down, first, that letter, N". or S., which is at the 0° 
point nearest to that end of the needle from which you are read- 
ing ; second, the number of degrees to which it points ; and, 
third, the letter E. or W. of the 90° point which is nearest to the 



TEE FIELD-WORK. 



113 



same end of the needle. Thus, in 
the figure, if when the sights were 
directed along a line (the north 
point of the compass being most 
distant from the observer) the 
north end of the needle was at the 
point A, the bearing of the line 
sighted on would be north 45° east ; 
if the end of the needle was at B, 
the bearing would be east ; if at 
0, S. 30° E. ; if at D, south ; if at 
E, S. 60° W. ; if at F, west; if at G, N. 60° W. ■ if at II, north. 




Fig. 150. 



/C 




181. We can now understand why W. is on the right hand of 
the compass-box and E. on the left. Let the direction from the 

center of the compass to the point 
B in the figure be required, and 
suppose the sights in the first place 
to be pointing in the direction of 
the needle, S. K, and the north 

w ]|), -g sight to be ahead. When the sights 

(and the circle to which they are 
fastened) have been turned so as to 
point in the direction of B, the 
point of the circle marked E. will 
have come round to the north end of the needle (since the 
needle remains immovable), and the reading will therefore be 
"east," as it should be. The effect on the reading is the same as 
if the needle had moved to the left the same distance which the 
sights have moved to the right, and the left side is therefore 
properly marked "east," and vice versa. So, too, if the bearing 
of the line to C be desired half-way between north and east — i. e., 
N. 45° E. ; when the sights and the circle have turned 45° to the 
right, the needle, really standing still, has apparently arrived at 
the point half-way between N. and E., i. e., N. 45° E. 

Some surveyors' compasses are marked the reverse of this, 
the E. on the right and the W. on the left. 



These letters must 



114: LAND-SURVEYING. 

then be reversed in the mind before the bearing is noted 
down. 

182. Reading with Vernier. When the needle does not point 
precisely to one of the division-marks on the circle, the fractional 
part of the smallest space is usually estimated by the eye, as has 
been explained. But this fractional part may be measured by the 
vernier as follows : Suppose the needle to point between N. 31° E. 
and N. 31£° E. Turn the tangent-screw which moves the com- 
pass-box till the smaller division (in this case 31°) has come round 
to the needle. The vernier will then indicate through what space 
the compass-box has moved, and therefore how much must be 
added to the reading of the needle. Suppose it indicates ten min- 
utes of a degree. Then the bearing is N". 31° 10' E. It is, how- 
ever, so difficult to move the vernier without disturbing the whole 
instrument, that this is seldom resorted to in practice. The chief 
use of the vernier is to set the instrument for running lines and 
making an allowance for the variation of the needle, as will be ex- 
plained in the proper place. A vernier arc is sometimes attached 
to one end of the needle and carried around by it. 

183. Practical Hints. Mark every station or spot at which the 
compass is set by driving a stake, or digging up a sod, or piling up 
stones, or otherwise, so that it can be found if any error or other 
cause makes it necessary to repeat the survey. 

Very often, when the line of which the bearing is required is a 
fence, etc., the compass can not be set upon it. In such cases, 
set the- compass so that its center is a foot or two from the line, 
and set the flag-staff at precisely the same distance from the line at 
the other end of it. The bearing of the flag-staff from the com- 
pass will be the same as that of the fence, the two lines being 
parallel. The distances should be measured on the real line. If 
more convenient, the compass may be set at some point on the line 
prolonged, or at some intermediate point of the line, "in line" 
between its extremities. 

In setting the compass level, it is more important to have it 
level crosswise of the sights than in their direction ; since, if it be 



TEE FIELD-WORK. 115' 

not so, on looking up or down hill through the upper part of one 
sight and the lower part of the other, the line of sight will not be 
parallel to the N. and S. or zero line on the compass, and an incor- 
rect bearing will therefore be obtained. 

The compass should not be leveled by the needle, as some books 
recommend — i. e., so leveled that the ends of the needle shall be 
at equal distances below the glass. The needle should be brought 
so originally by the maker, but, if so adjusted in the morning, it 
will not be so at noon, owing to the daily variation in the dip. If, 
then, the compass be leveled by it, the lines of sight will generally 
be more or less oblique, and therefore erroneous. If the needle 
touches the glass when the compass is leveled, balance it by sliding 
the coil of wire along it. 

The same end of the compass should always go ahead. The 
north end is preferable. The south end will then be nearest to 
the observer. Attention to this 'and to the caution in the next 
paragraph will prevent any confusion in the bearings. 

Always take the readings from the same end of the needle ; 
from the north end, if the north end of the compass goes ahead, 
and vice versa. This is necessary, because the two ends will not 
always cut opposite degrees. With this precaution, however, the 
angle of two meeting lines can be obtained correctly from either 
end, provided 'the same one is used in taking the bearings of both 
the lines. 

Guard against a very frequent source of error with beginners 
in reading from the wrong number of 
the two between which the needle points, FlG - 151, 

such as reading 34° for 26° in a case like rrXTTtninpn77777/ 
that in the figure. ^° 

Check the vibrations of the needle 
by gently raising it off the pivot so as 
to touch the glass, and letting it down again by the screw on 
the under side of the box. 

The compass should be smartly tapped after the needle has 
settled, to destroy the effect of any adhesion to the pivot or fric- 
tion of dust upon it. 

All iron, such as the chain, etc., must be kept at a distance 



116 LAND-SURVEYING. 

from the compass, or it will attract the needle, and cause it to 
deviate from its proper direction. 

The surveyor is sometimes troubled by the needle refusing to 
traverse and adhering to the glass of the compass after he has 
briskly wiped this off with a silk handkerchief, or it has been car- 
ried so as to rub against his clothes. The cause is the electricity 
excited by the friction. It is at once discharged by applying a wet 
finger to the glass. 

A compass should be carried with its face resting against the 
side of the surveyor, and one of the sights hooked over his arm. 

In distant surveys an extra center-pin should be carried (as it 
is very liable to injury, and its perfection is most essential), and 
also an extra needle. When two such are carried they should be 
placed so that the north pole of one rests against the south pole of 
the other. 

184. vVhen the magnetism of the needle is lessened or de- 
stroyed by time, it may be renewed as follows: Obtain two bar 
magnets. Provide a board with a hole to admit of the axis, so 
that its collar may fit fairly, and that the needle may rest flat on it 
without bearing at the center. Place the board before you with 
the north end of the needle to your right. Take a magnet in 
each hand, the left holding the north end of the bar, or that which 
has the mark across, downward, and the right holding the same 
mark upward. Bring the bars over the axis, about a foot above 
it, without approaching each other within two inches ; bring them 
down vertically on the needle (the marks as directed) about an 
inch on each side of its axis ; slide them outward to its ends with 
slight pressure ; raise them up ; bring them to their former posi- 
tion, and repeat this a number of times. 

185. Back-Sights. To test the accuracy of the bearing of a line 
taken at one end of it, set up the compass at the other end or point 
sighted to, and look back to a rod held at the first station or point 
where the compass had been placed originally. The reading of the 
needle should now be the same as before. 

If the position of the sights had been reversed, the reading 



THE FIELD-WORK. 117 

would be the Reverse Bearing ; a former bearing of N. 30° E. 
would then be S. 30° W., and so on. 



186. Local Attraction. If the back-sight does not agree with 
the first or forward sight, this latter must be taken over again. If 
the same difference is again found, this shows that there is local 
attraction at one of the stations — i. e., some influence, such as a 
mass of iron-ore, ferruginous rocks, etc., under the surface, which 
attracts the needle, and makes it deviate from its usual direction. 
Any high object, such as a house, a tree, etc., has been found 
to produce a similar effect. 

To discover at which station the attraction exists, set the com- 
pass at several intermediate points in the line which joins the two 
stations, and at points in the line prolonged, and take the bearing 
of the line at each of these points. The agreement of several of 
these bearings, taken at distant points, will prove their correctness. 
Otherwise, set the compass at a third station, sight to each of the 
two doubtful ones, and then from them back to this third station. 
This will show which is correct. 

When the difference occurs in a series of lines, such as around a 
field or along a road, proceed thus : Let C be the station at which 
the back-sight to B differs 
from the fore-sight from B to 
0. Since the back-sight from 
B to A is supposed to have 
agreed with the fore - sight 
from A to B, the local attrac- 



Fig. 152. 




Fig. 153. 




tion must be at C, and the forward 
bearing must be corrected by the dif- 
ference just found between the fore- 
and back-sights, adding or subtracting 
it, according to circumstances. An 
easy method is to draw a figure for the 
case, as in Fig. 153. In it, suppose 
the true bearing of B C, as given by 
a fore-sight from B to C, to be N. 40° 
E., but that there is local attraction 



118 LAND-SURVEYING. 

at C, so that the needle is drawn aside 10°, and points in the di- 
rection S'X' instead of SK The back-sight from C to B will 
then give a bearing of X. 50° E. ; a difference or correction for 
the next fore-sight of 10°. If the next fore-sight, from C to D, 
be IS". 70° E., this 10° must be subtracted from it, making the 
true fore-sight X. 60° E. 

A general rule may also be given. When the back-sight is 
greater than the fore-sight, as in this case, subtract the difference 
from the next fore-sight, if that course and the preceding one 
haye both their letters the same (as in this case, both being X. and 
E.), or both their letters different ; or add the difference if either 
the first or last letters of the two courses are different. When the 
hack-sight is less than the fore-sight, add the difference in the case 
in which it has just been directed to subtract it, and subtract it 
where it was before directed to add it. 

187. Angles of Deflection. When the compass indicates much 
local attraction, the difference between the directions of two meet- 
ing lines (or the "angle of deflection" of one from the other) can 
still be correctly measured by taking the difference of the bearings 
of the two lines, as observed at the same point. Eor the error 
caused by the local attraction, whatever it may be, affects both 
bearings equally, inasmuch as a "bearing" is the angle which a 
line makes with the direction of the needle, and that here remains 
fixed in some one direction, no matter what, during the taking of 
the two bearings. Thus, in Fig. 153, let the true bearing of B C — 
i. e., the angle which it makes with the line SX — be, as before, 
X. 40° E., and that of C D, X. 60° E. The true "'angle of deflec- 
tion " of these lines, or the angle B' C D, is therefore 20°. Xow, 
if local attraction at C causes the needle to point in the direction 
of S' W, 10° to the left of its proper direction, B C will bear X. 
50° E., and CD X. 70° E., and the difference of these bearings— 
i. e., the angle of deflection — will be the same as before. 

188. Angles between Courses. To determine the angle of de- 
flection of two courses meeting at any point, the following simple 
rules, the reasons of which will appear from the accompanying 
figures, are sufficient : 



THE FIELD-WORK. 



119 



Fig. 154. 




Case 1. When the first letters of the bearing are alike (i. e., 

both 1ST. or both S.), and the last letters also alike (i. e., both E. or 

both W.), take the difference of the 

bearings. Example : If A B bears N. 

30° E., and B C bears N. 10° E., the 

angle of deflection C B B' is 20°. 

Case 2. When the first letters are 

alike and the last letters different, take 

the sum of the bearings. Ex. : If A B 

bears N. 40° E. and B C bears N. 20° 

W., the angle C B B' is 60°. 

Case 3. When the first letters are 

different and the last letters alike, sub- 
tract the sum of the bearings from 180°. 
Ex. : If AB bears K 30° E. and B C 
bears S. 40° E., the angle C B B' is 110°. 
Case 4. When both the first and last 
letters are different, subtract the differ- 
ence of the bearings from 180°. Ex. : If 
A B bears S. 30° W. and B bears K 
70° E., the angle C B B' is 140°. 

If the angles included between the 
courses are desired, they will be at once 
found by reversing one bearing and then 

applying the above rules ; or by subtracting the results obtained 

as above from 180° ; or an analogous set of rules could be formed 

for them. 



Fig. 155. 




Fig. 156. 

N 

i 



"VF-- -<*&- 



Fig. 15Y. 



— E 





<$ S 



120 LAND-SURVEYING. 

189. To change Bearings. It is convenient in certain calcula- 
tions to suppose one of the lines of a survey to change its di- 
rection so as to become due north and south ; that is, to be- 
come a new meridian line. It is, then, necessary to determine 
what the bearings of the other lines will be, supposing them to 
change with it. The subject may be made plain by supposing 
the survey to be platted in the usual way, with the north up- 
permost, and the plat to be then turned around till the line to 
be changed is in the desired direction. The effect of this on 
the other lines will be readily seen. A general rule can also be 
formed : 

Take the difference between the original bearing of the side 
which becomes a meridian, and each of those bearings which have 
both their letters the same as it, or both different from it. The 
changed bearings of these lines retain the same letters as before, if 
they were originally greater than the original bearing of the new 
meridian line ; but, if they were less, they are thrown on the other 
side of the N". and S. line, and their last letters are changed, E. 
being put for W., and W. for E. 

Take the sum of the original bearing of the new meridian line, 
and each of those bearings which have one letter the same as one 
letter of the former bearing and one different. If this sum exceeds 
90°, this shows that the line is thrown on the other side of the 
east or west point, and the difference between this sum and 180° 
will be the new bearing, and the first letter will be changed, N. 
being put for S. and S. for N. 

Example : Let the bearings of the sides of a field be as follows : 
K 32° E. ; N. 80° E. ; S. 48° E. ; S. 18° W. ; N. 73^° W. ; North. 
Suppose the first side to become due north ; the changed bearings 
will then be as follows : North ; N. 48° E. ; S. 80° E. ; S. 14° E. ; 
S. 74^° W. ; N. 32° W. 

To apply the rule to the "North" course, as above, it must 
be called N. 0° W. ; and then, by the rule, 32° must be added 
to it. 

The true bearings can, of course, be obtained from the changed 
bearings by reversing the operation, taking the sum instead of the 
difference, and vice versa. 



THE FIELD-WORK. 



121 



190. Line-Surveying. This name may be given to surveys of 
lines, such as the windings of a brook, the curves of a road, etc., 
by way of distinction from Farm- Surveying, in which the lines 
surveyed inclose a space. 

To survey a brook, or any similar line, set the compass at or 
near one end of it, and take the bearing of an imaginary or visual 

Fig. 158. 




line running in the general average direction of the brook, such as 
A B in the figure. Measure this line, taking offsets to the various 
bends of the brook, as explained in Art. 97. Then set the com- 
pass at B, and take a back-sight to A, and, if they agree, take 
a fore-sight to C, and proceed as before, noting particularly the 
points where the line crosses the brook. 

To survey a road, take the bearings and lengths of the lines 

Fig. 159. 




which can be most conveniently measured in the road, and measure 
offsets' on each side to the outside of the road. 

When the line of a new road is surveyed, the bearings and 
lengths of the various portions of its intended center-line should be 
measured, and the distance which it runs through each man's land 
should be noted. Stones should be set in the ground at recorded 
distances from each angle of the line, or in each line prolonged 
a known distance, so as not to be disturbed in making the 
road. 

In surveying a wide river, one bank may be surveyed by the 
method just given, and points on the opposite banks, as trees, etc., 
may be fixed by the method of intersections founded on the fourth 
method of determining the position of a point. 



122 LAXD-SUR VETING. 

191. Checks by Intersecting Bearings. At each station at which 
the compass is set take bearings to some remarkable object, such as 
a church-steeple, a distant house, a high tree, etc. At least three 
bearings should be taken to each object to make it of any use, 
since two are necessary to determine it (by our fourth method), 
and, till thus determined, it can be no check. When the line is 
platted, by the methods to be explained hereafter, plat also the 
lines given by these bearings. If those taken to the same object 
from three different stations intersect in the same point, this 
proves that there has been no mistake in the survey or platting of 
those stations. 

If any bearing does not intersect a point fixed by previous bear- 
ings, ifc shows that there has been an error, either between the last 
station and one of those which fixed the point, or in the last bear- 
ing to the point. To discover which it was, plat the following 
line of the survey, and, at its extremity, set off the bearing from it 
to the point, and, if the line thus platted passes through the point, 
it proves that there was no error in the line, but only in the bearing 
to the point. If otherwise, the error was somewhere in the line 
between the stations from which the bearings to that point were 
taken. 

192. Keeping the Field-Notes. The simplest and easiest method 
for a beginner is to make a rough sketch of the survey by eye, and 
write down on the lines their bearings and lengths. 

An improvement on this is to actually lay down the precise 
bearings and lengths of the lines in the field-book in the manner to 
be explained in the section on Platting, Art. 209. 

193. A second method is to draw a straight line up the page of 
the field-book, and to write on it the bearings and lengths of the 
lines. The only advantage of this method is that the line will not 
run off the side of the page, as it is apt to do in the preceding 
method. 

194. A third method is to represent the line surveyed by a 
double column, as in Art. 84, which should be now referred to. 
The bearings are written obliquely up the columns. At the end of 



TEE FIELD-WORK. 



123 



each course its length is written in the column, and a line drawn 
across it. Dotted lines are drawn across the column at any inter- 
mediate measurement. Offsets are noted as explained in Art. 97. 
The intersection bearings, described in Art. 191, should be en- 
tered in the field-book before the bearings of the line, in order to 
avoid mistakes of platting in setting off the measured distances on 
the wrong line. 

195. A fourth method is to write the stations, bearings, and 
distances in three columns. This is compact, and has the advan- 
tage, when applied to farm-surveying, of presenting a form suitable 
for the subsequent calculations of content, but does not give facili- 
ties for noting offsets. 

Examples of these four methods are given in Art. 199, which 
contains the field-notes of the lines bounding a field. 

196. New York Canal-Maps. The following is a description of 
the original maps of the survey of the line of the New York Erie 
Canal, as published by the Canal Commissioners. The figure 
represents a portion of such a map, but, necessarily, with all its 
lines black, red and blue lines being used on the real map : 

"The Red Like described along the inner edge of the towing- 
path is the base-line, upon which all the measurements in the di- 

Fig. 160. 




rection of the length of the canal were made. The bearings refer 

to the magnetic meridian at the time of the survey. The lengths 

of the several portions are inserted at the end of each in chains and 

links. The offsets at each station are represented by red lines 

drawn across the canal in such a direction as to bisect the angles 
9 



124: LAXD-SURYEYIKG. 

formed by the two contiguous portions of the red or base line upon 
the towing-path. The intermediate offsets are set off at right 
angles to the base-line, and the distances on both are given from it 
in links. The intermediate offsets are represented by red dotted 
lines, and the distances to them upon the base-line are reckoned, 
in each case, from the last preceding station. The same is likewise 
done with the other distances upon the base-line ; those to the 
bridges being taken to the lines joining the nearest angles or corner 
posts of their abutments ; those to the lochs extending to the lines 
passing through the centers of the two nearest quoin-posts ; and 
those to the aqueducts to the faces of their abutments. The space 
inclosed by the Blue Lixes represents the portion embraced with- 
in the limits of the survey as belonging to the State ; and the 
names of the adjoining proprietors are given as they stood at the 
time of executing the survey. The distances are projected upon a 
scale of two chains to the inch." 

187. Farm-Surveying. A farm or field or other space included 
within known lines is usually surveyed by the compass thus : Begin 
by walking around the boundary-lines and setting stakes at all the 
corners, which the flag-man should specially note, so that he may 
readily find them again. Then set the compass at any corner, and 
send the flag-man to the next corner. Take the bearing of the 
bounding-line running from corner to corner, which is usually a 
fence. Measure its length, taking offsets if necessary. Kote 
where any other fence, or road, or other line crosses or meets it, 
and take their bearings. Take the compass to the end of this first 
bounding-line ; sight back, and, if the back-sight agrees, take the 
bearing and distance of the next bounding-line : and so proceed till 
you have got back to the point of starting. 

198. "Where speed is more important than accuracy in a sur- 
vey, whether of a line or a farm, the compass need be set only at 
every other station, taking a forward sight from the first station to 
the second ; then, setting the compass at the third station, taking 
a back-sight to the second station (but with the north point of the 
compass always ahead), and a fore-sight to the fourth : then going 
to the fifth, and so on. This is, however, not to be recommended. 



THE FIELD-WORK. 



125 



199. Field-Notes. 

The field-notes of a 
farm-survey may be 
kept by any of the 
methods which have 
been described with 
reference to a line- 
survey. Below are 
given the field-notes 
of the same field re- 
corded by each of 
the methods. 



Second Method. 
0(1) 




©(5) 



G(4) 



^ 



CO 



Third Method. 
3-23 

O 
lO 

£; 

-(5)- 
3-55 



m 
-(4)- 

2-22 



0(3) 



H 



0(2) 



Ee] 



O(l) 



m 
-(3)- 
1-29 



-(2)- 
270 



(i)- 



Fourth Method. 



STATION?. 


BEARINGS. 


DISTANCES. 


1 

2 

I 

5 


N. 35° E. 
K 83£° E. 
S. 57° E. 
S. 34J° W. 
N. 56|° W. 


2-70 
1-29 

2-22 
3-55 
3-23 




* In the " third method ° the bearings should be written obliquely upward, 
directed in Art. 194, but are not so printed here, from typographical difficulties. 



126 LAND-SURVEYING. 

200. The field-notes of a field in which offsets occur may be 
most easily recorded by the third method, as in Fig. 162. 

When the field-notes are recorded by the fourth method, the 
offsets may be kept in a separate table, in which the first column 
will contain the stations from which the measurements are made, 
the second column the distances at which they occur, the third 
column the lengths of the offsets, and the fourth column the side 
of the line, "right" or "left," on which they lie. 

For calculation, four more columns may be added to the table, 
containing the intervals between the offsets, the sums of the ad- 
joining pairs, and the products of the numbers in the two preced- 
ing columns, separated into right and left, one being additive to 
the field, and the other sub tractive. 

201. Tests of Accuracy. 1. The check of intersections de- 
scribed in Art. 191 may be employed to great advantage when 
some conspicuous object near the center of the farm can be seen 
from most of its corners. 

2. When the survey is platted, if the last course meets the 
starting-point, it proves the work, and the survey is then said to 
"close." 

3. Diagonal lines running from corner to corner of the farm, 
like the " proof-lines " in chain-surveying, may be measured and 
their bearings taken. When these are laid down on the plat, their 
meeting the points to which they had been measured proves the 
work. 

4. The only certain and precise test is, however, that by "lati- 
tudes and departures." 

202. Method of Radiation. A field may be surveyed from one 
station, either within it or without it, by taking the bearings and 
the distances from that point to each of the corners of the field. 
These corners are then " determined " by the third method, Art. 
5. This modification of that method is called the Method of 
Radiation. All our preceding surveys with the compass have been 
by the Method of Progression. 

The compass may be set at one corner of the field, or at a point 



THE FIELD-WORK. 127 

in one of its sides, and the same method of radiation em- 
ployed. 

This method is seldom used, however, since, unlike the method 
of progression, its operations are not checks upon each other. 

203. Method of Intersection. A field may also be surveyed by 
measuring a base-line, either within it or without it, setting the 
compass at each end of the base-line, and taking from each end the 
bearings of each corner of the field, which will then be fixed and 
determined by the fourth method, Art. 6. This mode of survey- 
ing is the Method of Intersections, noticed in Art. 166. 

204. Running out Old Lines. The original surveys of lands in 
the older States of the American Union were exceedingly deficient 
in precision. This arose from two principal causes : the small 
value of land at the period of these surveys, and the want of skill 
in the surveyors. The effect at the present day is frequent dissat- 
isfaction and litigation. Lots sometimes contain more acres than 
they were sold for, and sometimes less. Lines which are straight 
in the deed and on the map are found to be crooked on the ground. 
The recorded surveys of two adjoining farms often make one over- 
lap the other, or leave a gore between them. The most difficult 
and delicate duty of the land-surveyor is to run out these old 
boundary-lines. In such cases, his first business is to find monu- 
ments, stones, marked trees, stumps, or any other old " corners " 
or landmarks. These are his starting-points. The owners whose 
lands join at these corners should agree on them. Old fences must 
generally be accepted by right of possession, though such questions 
belong rather to the lawyer than to the surveyor.* His business is 
to mark out on the ground the lines given in the deed. When the 
bounds are given by compass-bearings, the surveyor must be re- 
minded that these bearings are very far from being the same now 
as originally, having been changing every year. The method of 

* " In the description of land conveyed, the rule is that known and fixed monu- 
ments control courses and distances. So the certainty of metes and bounds will in- 
clude and pass all the lands within them, though they vary from th,e given quantity 
expressed in the deed. In New York, to remove, deface, or alter landmarks 
maliciously is an indictable offense." — KenVs Commentaries, IV, 515. 



128 



LAXD-SUR YEYING. 



determining this important change, and of making the proper 
allowance, will be found under " Declination of the Magnetic 
Needle." 

PLATTING THE SURVEY. 

205. The platting of a survey made with the compass consists 
in drawing on paper the lines and the angles which have been 
measured on the ground. The angles are laid off and the lines are 
drawn "to scale," as has been explained in Chapter I. 

206. Platting Bearings. Since "bearings" taken with the 
compass are the angles which the various lines make with the 
magnetic meridian, or the direction of the compass-needle, which, 
as we have seen, remains always (approximately) parallel to itself, 
it is necessary to draw these meridians through each station before 
laying off the angles of the bearings. 



The T-square is the most convenient instrument for this purpose. The 
paper on which the plat is to be made is fastened on the board so that 
the intended direction of the north and south line may be parallel to one 
of the sides of the board. The inner side of the stock of the T-square being 
pressed against one of the other sides of the board and slid along, the edge 
of the long blade of the square will always be parallel to itself and to the 
first-named side of the board, and will thus represent the meridian passing 
through any station. 

If a straight-edged drawing-board or table can not be procured, nail 

down on a table of any shape a 



Fig. 163. 




straight-edged ruler, and slide along 
against it the outside of the stock 
of a T-square, one side of the stock 
being flush with the blade. 

A parallel ruler may also be 
used, one part of it being screwed 
down to the board in the proper 
position. 

If none of these means are at 
hand, approximately parallel merid- 
ians may be drawn by the edge? of 
a common ruler at distances apart 
equal to its width, and the diameter 
of the protractor made parallel to 
them by measuring equal distances 
between it and them. 



PLATTING TEE SURVEY. 



129 



Fio. 164. 



207. To plat a survey with these instruments, mark with a fine 
point inclosed in a circle a convenient spot in the paper to repre- 
sent the first sta- 
tion, 1 in the figure. 
Its place must be so 
chosen that the plat 
may not "run off" 
the paper. With 
the T-square draw a 
meridian through 
it. The top of the 
paper is usually, 
though not necessa- 
rily, called north. 
With the protractor 
lay off: the angle of 
the first bearing. 
Set off the length of 
the first line to the 
desired scale from 
1 to 2. The line 1----2 represents the first course. 

Through 2 draw another meridian, lay off the angle of the sec- 
ond course, and set off the length of this course from 2 to 3. 

Proceed in like manner for each course. When the last course 
is platted, it should end precisely at the starting-point, as the sur- 
vey did, if it were a closed survey, as of a field. If the plat does 
not "close" or "come together," it shows some error or inaccu- 
racy either in the original survey, if that have not been "tested" 
by latitudes and departures, or in the work of platting. The plat 
here given is the same as that of Fig. 161. 

This manner of laying down the directions of lines by the angles which 
they make with a meridian line has a great advantage, in both accuracy and 
rapidity, over the method of platting lines by the angles which each makes 
with the line which comes before it. In the latter method, any error in the 
direction of one line makes all that follow it also wrong in their directions. 
In the former, the direction of each line is independent of the preceding line, 
though its position would be changed by a previous error. 

Instead of drawing a meridian through each station, sometimes only one 




130 



LAXD- SUE TEYING. 



is drawn, near the middle of the sheet, and all the bearings of the survey are 
laid off from some one point of it, as shown in the figure, and numbered to 
correspond with the stations from which these bearings were taken. The 
circular protractor is convenient for this. They are then transferred to the 
places where they are wanted by a triangle or other parallel ruler. Fig. 
165 represents the same field platted by this method. 

A semicircular protractor is sometimes attached to the stock end of the 
T-square so that its blade may be set at any desired angle with the meridian, 
and any bearing be thus protracted without drawing a meridian. It has 
some inconveniences. 

The compass itself may be used to plat bearings. For this pur- 
pose it must be at- 
tached to a square 
board so that the N 
and S line of the 
compass-box may be 
parallel to two oppo- 
site edges of the 
board. This is placed 
on the paper, and 
the box is turned till 
the needle points as 
it did when the first 
bearing was taken. 
Then a line drawn 
by one edge of the 
board will be in a 

proper direction. Mark off its length, and plat the next and the 

succeeding bearings in the same manner. 

208. When the plat of a survey does not "close," it may be 
corrected as follows : Let A B C D E be the boundary-lines platted 
according to the given bearings and distances, and suppose that the 
last course comes to E instead of ending at A, as it should. Sup- 
pose also that there is no reason to suspect any single great error, 
and that no one of the lines was measured over very rough ground, 
or was specially uncertain in its direction when observed. The 
inaccuracy must then be distributed among all the lines in propor- 
tion to their length. Each point in the figure, B, C, I>, E, must 




S 5 



PLATTING TEE SURVEY. 



131 



Fig. 166. 




^C 



-* 



be moved in a direction parallel to E A by a certain distance which 

is obtained thus : Multiply 

the distance E A by the 

distance A B, and divide 

by the sum of all the 

courses. The quotient will 

be the distance B B'. To 

get CC, multiply E A by 

A B + B C, and divide the 

product by the same sum 

of all the courses. To get 

D D', multiply E A by 

AB-f BC-f-CD, and divide as before. So for any course, 

multiply by the sum of the lengths of that course and of all 

those preceding it, and divide as before. Join the points thus 

obtained, and the closed polygon AB'C'D'A will thus be formed, 

and will be the most probable plat of the given survey.* 

The method of latitudes and departures, to be explained here- 
after, is, however, the best for effecting this object. 



B ? 



209. Field Platting. 



Fig. 167. 



It is sometimes desirable to plat the courses of a 
survey in the field as soon as they 
are taken, as was mentioned in Art. 
192 under the head of " Keeping the 
Field-Notes." One method of do- 
ing this is to have the paper of the 
field-book ruled with parallel lines 
at unequal distances apart, and to 
use a rectangular protractor (which 
may be made of Bristol-board or 
other stout drawing-paper) with 
lines ruled across it at equal dis- 
tances of some fraction of an inch. A bearing having been taken and noted, 
the protractor is laid on the paper and its center placed at the station where 
the bearing is to be laid off. It is then turned till one of its cross-lines co- 
incides with some one of the lines on the paper, which represent east and 
west lines, The long side of the protractor will then be on a meridian, and 
the proper angle (40° in the figure) can be at once marked off. The length 
of the course can also be set off by the equal spaces between the cross-lines, 
letting each space represent any convenient number of links. 




* This was demonstrated by Dr. Bowditch in No. 4 of " The Analyst." 



132 



LAXD-SUR YEYIXG. 



210. A common rectangular protractor without any cross-lines, or a 

semicircular one, can also be used 
for the same purpose. The parallel 
lines on the paper (which, in this 
method, may be equidistant, as in 
common ruled writing-paper) will 
now represent meridians. Place 
the center of the protractor on the 
meridian nearest to the station at 
which the angle is to be laid off, and 
turn it till the given number of 
degrees is cut by the meridian. 
Slide the protractor up or down the 
meridian (which must continue to 
pass through the center and the 
proper degree) till its edge passes 

through the station, and then draw by this edge a line, which will have 

the bearing required. 




211. Paper ruled into squares (as are sometimes the right-hand pages of 
surveyors 1 field-books) may be used 
for platting bearings in the field. 
The lines running up the page 
may be called north and south 
lines, and those running across 
the page will then be east and 
west lines. Any course of the 
survey will be the hypotenuse of 
a right-angled triangle, and the 
ratio of its other two sides will 
determine the angle. Thus, if the 
ratio of the two sides of the right- 
angled triangle, of which the line 
AB in the figure is the hypote- 
nuse, is 1, that line makes an angle 
of 45° with the meridian. If the 
ratio of the long to the short side 
of the right-angled triangle, of 

which the line A C is the hypotenuse, is 4 to 1, the line A C makes an angle 
of 14° with the meridian. The line A I), the hypotenuse of an equal trian- 
gle which has its long side lying east and west, makes likewise an angle of 
14° with that side, and therefore makes an angle of 76° with the meridian. 



! ! 






Fig. 169 












cl 1 .1 


3 ; 






A 












1 








/ 






i 

i 




// 


-E 


1 


/ 








i 

i 




1/ 










! 


! / 


t 


!/! 






! 1// 




/ 


1 




I 


// 


/ 

/ 


\J, 


tc 




^T 




AT 





212. With a Paper Protractor. Engraved paper protractors may be 
obtained from the instrument-makers, and are very convenient. A circle of 
large size, divided into degrees and quarters, is engraved on copper, and im- 
pressions from it are taken on drawing-paper. The divisions are not num- 



PLATTING THE SURVEY. 



133 




bered. Draw a straight line to represent a meridian through the center of 
the circle in any convenient direction. Number the degrees from 0° to 90° 
each way from the ends of this 
meridian, as on the compass-plate. 
The protractor is now ready for 
use. Choose a convenient point 
for the first station. Suppose the 
first bearing to be N. 30° E. The 
line passing through the center of 
the circle and through the oppo- 
site points JST. 30° E. and S. 30° 
W. has the bearing required. But 
it does not pass through the sta- 
tion 1. Transfer it thither by 
drawing through station 1 a line 
parallel to it, which will be the 

course required, its proper length being set off on it from 1 to 2. Now, 
suppose the bearing from 2 to be S. 60° E. Draw through 2 a line parallel 
to the line passing through the center of the circle and through the opposite 
points S. 60° E. and N. 60° W., and it will be the line desired. On it set off 
the proper length from 2 to 3, and so proceed. 

When the plat is completed, the engraved sheet is laid on a clean one and 
the stations "pricked through," and the points thus obtained on the clean 
sheet are connected by straight lines. The penciled plat is then rubbed off 
from the engraved sheet, which can be used for a great number of plats. 

If the central circle be cut out, the plat, if not too large, can be made 
directly on the paper where it is to remain. 

The surveyor can make such a paper protractor for himself with great 
ease by means of the Table of Chords at the end of this volume, the use of 
which is explained in Art. 215. The engraved ones may have shrunk after 
being printed. 

Such a circle is sometimes drawn on the map itself. This will be particu- 
larly convenient if the bearings of any lines on the map not taken on the 
ground are likely to be required. If the map be very long, more than one 
may be needed. 

213. Drawing-Board Protractor. Such a divided circle as has just 
been described, or a circular protractor, may be placed on a drawing-board 
near its center, and so that its 0° and 90° lines are parallel to the sides of the 
drawing-board. Lines are then to be drawn through the center and oppo- 
site divisions by a ruler long enough to reach the edges of the drawing-board 
on which they are to be cut in and numbered. The drawing-board thus be- 
comes, in fact, a double rectangular protractor. A strip of white paper may 
have previously been pasted on the edges, or a narrow strip of white wood 
inlaid. When this is to be used for plattiug, a sheet of paper is put on the 
board as usual, and lines are drawn by a ruler laid across the 0° points and 
the 90° points, and the center of the circle is at once found, and should be 
marked 0. The bearings are then platted as in the last method. 




I3i land-survetixg: 

214. With a Scale of Chords. On the plane scale contained in cases of 
mathematical drawing instruments will be found a series of divisions num- 
bered from to 90, and marked C H or C. This is a 
Fig. 171. scale of chords, and gives the lengths of the chords of 

any arc for a radius equal in length to the chord of 60° 
on the scale. To lay off an angle with this scale, as, 
for example, to draw a line making at A an angle of 
40° with A B, take, in the dividers, the distances from 
to 60 on the scale of chords ; with this for radius and 
A for center, describe an indefinite arc D. Take the 
distance from to 40 on the same scale, and set it off 
on the arc as a chord from C to some point D. Join 
A D and prolong it. B A E is the angle required. 

The sector, Fig. 29, supplies a modification of this 
method sometimes more convenient. On each of its legs is a scale marked 
or C H. Open it at pleasure ; extend the compass from 60 to 60, one on 
each leg, and with this radius describe an arc. Then extend the compasses 
from 40 to 40, and the distance will be the chord of 40° to that radius. 
It can be set off as above. 

The smallness of the scale renders the method with a scale of chords 
practically deficient in exactness, but it serves to illustrate the next and lest 
method. 



215. With a Table of Chords. At the end of this volume will be found 
a table of the lengths of the chords of arcs for every degree and minute of 
the quadrant calculated for a radius equal to 1. 

To use it, take in the compasses one inch, one foot, or any other conven- 
ient distance (the longer the better), divided into tenths and hundredths by a 
diagonal scale or otherwise. "With this as radius describe an arc as in the 
last case. Find in the table of chords the length of the chord of the desired 
angle. Take it from the scale just used to the nearest decimal part which 
the scale will give. Set it off as a chord, as in the last figure, and join the 
point thus obtained to the starting-point. This gives the angle desired. 

The superiority of this method to that which employs a protractor is due 
to the greater precision with which a straight line can be divided than can a 
circle. 

A slight modification of this method is to take in the compasses ten equal 
parts of any convenient length, inches, half inches, quarter inches, or any 
other at hand, and with this radius describe an arc as before, and set off a 
chord ten times as great as the one found in the table — i. e.. imagine the 
decimal-point moved one place to the right. 

If the radius be 100 or 1,000 equal parts, imagine the decimal-point moved 
two or three places to the right. 

Whatever radius may be taken or given, the product of that radius into a 
chord of the table will give the chord for that radius. 

This gives an easy and exact method of getting a right angle by describ- 
ing an arc with a radius of 1, and setting off a chord equal to 1*4142. 



PLATTING THE SURVEY. 135 

If the angle to be constructed is more than 90°, construct on the other 
side of the given point upon the given line prolonged an angle equal to what 
the given angle wants of 180° — i. e., its supplement, in the language of 
trigonometry. 

This same table gives the means of measuring any angle. With the an- 
gular point for a center, and 1 or 10 for a radius, describe an arc. Measure 
the length of the chord of the arc between the legs of the angle, find this 
length in the table, and the angle corresponding to it is the one desired. 

This table will also serve to find the natural sine or cosine of any angle. 
Multiply the given angle by two ; find in the table the chord of this double 
angle ; and half of this chord will be the natural sine required. For the 
chord of any angle is equal to twice the sine of half the angle. To find the 
cosine, proceed as above, with the angle which, added to the given angle, 
would make 90°. 

Another use of this table is to inscribe regular polygons in a circle by set- 
ting off the chords of the arcs which their sides subtend. 

Still another use is to divide an arc or angle into any number of equal 
parts by setting off the fractional arc or angle. 

216. With a Table of Natural Sines. In the absence of a table of 
chords, heretofore rare, a table of natural sines, which can be found any- 
where, may be used as a less convenient substitute. Since the chord of any 
angle equals twice the sine of half the angle, divide the given angle by two : 
find in the table the natural sine of this half angle ; double it, and the prod- 
uct is the chord of the whole angle. This can then be used precisely as 
was the chord in the preceding article. 

An ingenious modification of this method has been much used. Describe 
an arc from the given point as center, as in the last two articles, but with a 
radius of five equal parts. Take from a table the length of the natural sine 
of half the given angle to a radius of ten. Set off this length as a chord on 
the arc just described, and join the point thus obtained to the given point. 

The reason of this is apparent from the figure. D E is the sine of half 
the angle B A C to a radius of ten equal parts, and B C is the chord directed 
to be set off to a radius of five equal parts. B C is 
equal to DE, forBO = 2'BF by trigonometry, and Fig. 112. 

D E = 2- B F by similar triangles ; hence B = D E. \ / 



217. By Latitudes and Departures. When 
the latitudes and departures of a survey have 
been obtained and corrected, either to test its 
accuracy or to obtain its content, they afford 
the easiest and best means of platting it. The 
description of this method will be given in Art. 
246. 



136 LAND-SUR YEYIXG. 

COPYING PLATS. 

218. The plat of a survey necessarily has many lines of con- 
struction drawn upon it which are not needed in the finished map. 
These lines and the marks of instruments so disfigure the paper 
that a fair copy of the plat is usually made before the map is 
finished. The various methods of copying plats, etc., whether 
on the same scale, or reduced, or enlarged, will therefore now be 
described. 

219. Stretching the Paper. If the map is to be colored, the 
paper must first be wetted and stretched, or the application of the 
wet colors will cause its surface to swell or blister and become un- 
even. Therefore, with a soft sponge and clean water, wet the back 
of the paper, working from the center outward in all directions. 
The " water-mark " reads correctly only when looked at from the 
front side, which it thus distinguishes. AVhen the paper is thor- 
oughly wet and thus greatly expanded, glue its edges to the draw- 
ing-board for half an inch in width, turning them up against a 
ruler, passing the glue along them, and then turning them down 
and pressing them with the ruler. Some prefer gluing down oppo- 
site edges in succession, and others adjoining edges. The paper 
must be moderately stretched smooth during the process. Hot 
glue is best. Paste or gum may be used, if the paper be kept wet 
by a damp cloth, so that the edges may dry first. "Mouth-glue " 
may be used by rubbing it (moistened in the mouth or in boiling 
water) along the turned-up edges, aud then rubbiug them dry by 
an ivory folder, a piece of dry paper being interposed. As this is a 
slower process, the middle of each side should first be fastened 
down, then the four angles, and lastly the intermediate portions. 
When the paper becomes dry, the creases and puckerings will have 
disappeared, and it will be as smooth and tight as a drum-head. 

220. Copying by Tracing. Fix a large pane of clear glass in a 
frame so that it can be supported at any angle before a window, or. 
at night, in front of a lamp. Place the plat to be copied on this 
glass, and the clean paper upon it. Connect them by pins, etc. 
Trace all the desired lines of the original with a sharp pencil as 



COPYING PLATS. 137 

lightly as they can be easily seen. Take care that the paper does 
not slip. If the plat is larger than the glass, copy its parts succes- 
sively, being very careful to fix each part in its true relative posi- 
tion. Ink the lines with India ink, making them very fine and 
pale if the map is to be afterward colored. 

221. Copying on Tracing-Paper. A thin transparent paper is 
prepared expressly for the purpose of making copies of maps and 
drawings, but it is too delicate for much handling. It may be pre- 
pared by soaking tissue-paper in a mixture of turpentine and Can- 
ada balsam or balsam of fir (two parts of the former to one of the 
latter), and drying very slowly. Cold-drawn linseed-oil will answer 
tolerably, the sheets being hung up for some weeks to dry. Linen 
is also similarly prepared, and sold under the name of " vellum 
tracing-paper." It is less transparent than the tracing-paper, but 
is very strong and durable. Both of these are used rather for pre- 
serving duplicates than for finished maps. 

222. Copying by Photography. This may be used for copying 
drawings, and is especially applicable when the drawings are to be 
very much reduced in size. 

223. Copying by Blue Prints. Dissolve one ounce of ferricya- 
nide of potassium in ten ounces of pure water. Also dissolve two 
ounces of ammonia citrate of iron in ten ounces of water. Mix the 
two solutions in a cup, and with a brush cover the surface of the 
paper on which the print is to be made with the mixture. 

The surface should be thoroughly covered, but no more of the 
mixture should be applied than the paper will take up. The paper 
should become limp and moist but not wet. The work should be 
done in a room lighted with a lamp, and when the paper is dry it 
should be kept in a dark place. 

To make a blue-print copy, a tracing of the drawing should 
first be made. Put the tracing over a sheet of the prepared paper 
and a sheet of glass over the tracing, in order to keep the tracing 
in contact with the prepared paper. Expose the paper to the sun- 
light, with the glass toward the sun, until the lines of the drawing 
are plainly seen on the prepared paper. Wash the paper until the 



133 LAND-SUR VEYING. 

water running off is no longer colored yellow. When dried, the 
lines of the drawing will be white upon a blue ground. The pre- 
pared paper for blue prints can be bought of dealers in engineers' 
supplies. 

There are several similar methods of making prints, differing in 
the chemicals used, and in the color of the lines and background. 

224. Copying by Transfer-Paper. This is thin paper, one side of which 
is rubbed with black-lead, etc., smoothly spread by cotton. It is laid on the 
clean paper, the blackened side downward, and the plat is placed upon it. 
All the lines of the plat are then gone over with moderate pressure by a 
blunt point, such as the eye-end of a small needle. A faint tracing of these 
lines will then be found on the clean paper, and can be inked at leisure. If 
the original can not be thus treated, it may first be copied on tracing-paper, 
and this copy be thus transferred. If the transfer-paper be prepared by 
rubbing it with lampblack ground up with hard soap, its lines will be in- 
effaceable. It is then called " Camp-paper." 

225. Copying by Punctures. Fix the clean paper on a drawing-board 
and the plat over it. Prepare a fine needle with a sealing-wax head. Hold 
it very truly perpendicular to the board, and prick through every angle of 
the plat, and every corner and intersection of its other lines, such as houses, 
fences, etc., or at least the two ends of every line. For circles, the center 
and one point of the circumference are sufficient. For irregular curves, such 
as rivers, etc., enough points must be pricked to indicate all their sinuosities. 
AVork with system, finishing up one strip at a time, so as not to omit any 
necessary points nor to prick through any twice, though the latter is safer. 
When completed, remove the plat. The copy will present a wilderuess of 
fine points. Select those which determine the leading lines, and then the 
rest will be easily recognized. A beginner should first pencil the lines lightly, 
and then ink them. An experienced draughtsman will omit the penciling. 
Two or three copies may be thus pricked through at once. The holes in the 
original plat may be made nearly invisible by rubbing them on the back of 
the sheet with a paper-folder, or the thumb-nail. 

226. Copying by Intersections. Draw a line on the clean paper equal 
in length to some important line of the original. Two starting-points are 
thus obtained. Take in the dividers the distance from one end of the line 
on the original to a third point. From the corresponding end on the copy, 
describe an arc with this distance for radius and about where the point will 
come. Take the distance on the original from the other end of the line to 
the point, and describe a corresponding arc on the copy to intersect the for- 
mer arc in a point which will be that desired. The principle of the opera- 
tion is that of our " First Method" (Art. 3). Two pairs of dividers may be 
used, as explained in Art. 82. " Triangular compasses," having three legs, 
are used by fixing two of their legs on the two given points of the original, 



COPYING PLATS. 139 

and the third leg on the point to be copied, and then transferring them to 
the copy. All the points of the original can thus be accurately reproduced. 
The operation is, however, very slow. Only the chief points of a plat may 
be thus transferred, and the details filled in by the following method : 

227. Copying by Squares. On the original plat draw a series of par- 
allel and equidistant lines. The T-square does this most readily. Draw a 
similar series at right angles to these. The plat will then be covered with 
squares, as in Fig. 43. On the clean paper draw a similar series of squares. 
The important points may now be fixed as in the last article, and the rest 
copied by eye, all the points in each square of the original being properly 
placed in the corresponding square of the copy, noticing whether they are 
near the top or bottom of each square, on its right or left side, etc. This 
method is rapid, and in skillful hands quite accurate. 

Instead of drawing lines on the original, a sheet of transparent paper 
containing them may be placed over it; or an open frame with threads 
stretched across it at equal distances and at right angles. 

This method supplies a transition to the Reduction and Enlargement of 
plats in any desired ratio ; under which head Copying by the Pantagraph 
and Camera Lucida will be noticed. 

228. Reducing by Squares. Begin, as in the preceding article, by draw- 
ing squares on the original, or placing them over it. Then on the clean 
paper draw a similar set of squares, but with their sides one half, one third, 
etc. (according to the desired reduction), of those of the original plat. Then 
proceed as before to copy into each small square all the points and lines 
found in the large square of the plat in their true positions relative to the 
sides and corners of the square, observing to reduce each distance, by eye, or 
as directed in the following article, in the given ratio. 

229. Reducing by Proportional Scales. Many graphical methods of 
finding the proportionate length of the 

copy, of any line of the original, may FlG - 1 ' 73 - 

be used. The " angle of reduction " is \C__ 

constructed thus : Draw any line A B. ,^\\ \ 

With it for radius and A for center, "E_.'\\\ \ \ 

describe an indefinite arc. With B for 



^111 



center and a radius equal to one half, 

one third, etc., of A B according to the sf<*\\ x \ \ \ \ ^ ' * \ \ > \ \ ^ 

desired reduction, describe another arc ^s\\ \ \ \ \ 1 i i i j i i ! 1 i 1 1 i i \ 

intersecting the former arc in 0. Join A D B 

A 0. From A as center describe a 

series of arcs. Now, to reduce any distance, take it in the dividers, and set 

it off from A on A B, as to D. Then the distance from D to E, the other 

end of the arc passing through D, will be the proportionate length to be set 

off on the copy, in the manner directed in Art. 226. 

The sector, or " compass of proportion," described in Art. 50, presents 
such an "angle of reduction," always ready to be used in this manner. 
10 




i4: LAND-SUE VEYING. 

Fig. 174. The " angle of reduction " may be simplified thus : 

Draw a line, A B, .parallel to one side of the draw- 
ing-board, and another, B C, at right angles to it, and 
one half, etc., of it, as desired. Join A C. Then let 
A D be the distance required to be reduced. Apply 
a T-square so as to pass through D. It will meet A 
in some point E, and D E will be the reduced length 
required. 

. Another arrangement for the same object is shown 
in Fig. 175. Draw two lines, A B, A C, at any angle, 
and describe a series of arcs from their intersection, 
A, as in the figure. Suppose the reduced scale is to 

be half the original scale. Divide the outermost arc into 

three equal parts, and draw a line from A to one of the 

points of division, as D. Then each arc will be divided 

into parts, one of which is twice the other. Take any 

distance on the original scale, and find by trial which of 

the arcs on the right-hand side of the figure it corre- 
sponds to. The other part of that arc will be half of it, 

as desired. 

" Proportional compasses," being properly set, reduce 

lines in any desired ratio. A simple form of Jhem, 

known as " wholes and halves," is often useful. It con- 
sists of two slender bars, pointed at each end, and united 

by a pivot which is twice as far from one pair of the 

points as from the other pair. The long ends being set to any distance, the 

short ends will give precisely half that distance. 

230. Reducing by a Pantagraph. This instrument consists of two 
long and two short rulers, connected so as to form a parallelogram, and capa- 
ble of being so adjusted that when a tracing- point attached to it is moved 
over the lines of a map, etc., a pencil attached to another part of it will mark 
on paper a precise copy, reduced on any scale desired. It is made in various 
forms. It is troublesome to use, though rapid in its work. 

231. Reducing by a Camera Lucida. This is used in the Coast Sur- 
vey Office. It can not reduce smaller than one fourth, without losing dis- 
tinctness, and is very trying to the eyes. Squares drawn on the original are 
brought to apparently coincide with squares on the reduction, and the details 
are then filled in with the pencil, as seen through the prism of the instrument. 

232. Enlarging Plats. Plats may be enlarged by the principal meth- 
ods which have been given for reducing them, but this should be done as 
seldom as possible, since every inaccuracy in the original becomes magnified 
in the copy. It is better to make a new plat from the original data. 

233. Conventional Signs. Various conventional signs or marks have 
been adopted, more or less generally, to represent on maps the inequalities of 




COPYING PLATS. 141 

the surface of the ground, its different kinds of culture or natural products, 
and to objects upon it, so as not to encumber and disfigure it with much 
writing or many descriptive legends. This is the purpose of. what is called 
Topographical Mapping. (See Part III, Topography.) 

234. Orientation. The map is usually so drawn that the top of the 
paper may represent the north. A meridian line should also be drawn, both 
true and magnetic, as in Fig. 186. The number of degrees and minutes in 
the variation, if known, should also be placed between the two north points. 
Sometimes a compass-star is drawn and made very ornamental. 

235. Lettering. The style in which this is done very much affects the 
general appearance of the map. The young surveyor should give it much 
attention and careful practice. It must all be in imitation of the best printed 
models. No writing, however beautiful, is admissible. The usual letters are 
the ordinary KOMAN CAPITALS, Small Roman, ITALIC CAPITALS, 
Small Italic, and GOTHIC OR EGYPTIAN. This last, when well 
done, is very effective. For the titles of maps, various fancy letters may be 
used. For very large letters, those formed only of the shades of the letters 
regarded as blocks (the body being rubbed out after being penciled as a guide 
to the placing of the shades) are most easily made to look well. The simplest 
lettering is generally the best. The sizes of the names of places, etc., should 
be proportional to their importance. Elaborate tables for various scales have 
been published. It is better to make the letters too small than too large. 
They should not be crowded. Pencil-lines should always be ruled as guides. 
The lettering should be in lines parallel to the bottom of the map, except the 
names of rivers, roads, etc., whose general course should be followed. 

236. Borders. The Border may be a single heavy line, inclosing the 
map in a rectangle, or such a line may be relieved by a finer line drawn par- 
allel and near to it. Time should not be wasted in ornamenting the border. 
The simplest is the best. 

237. Joining Paper. If the map is larger than the sheets of paper at 
hand, they should be joined with a feather-edge, by proceeding thus : Cut, 
with a knife guided by a ruler, about one third through the thickness of the 
paper, and tear off, on the under side, a strip of the remaining thickness, so 
as to leave a thin, sharp edge. Treat the other sheet in the same way on the 
other side of it. When these two feather-edges are then put together (with 
paste, glue, or varnish), they will make a neat and strong joint. The sheet 
which rests upon the other must be on the right-hand side, if the sheets are 
joined lengthwise, or below if they are joined in that direction, so that the 
thickness of the edge may not cast a shadow when properly placed as to the 
light. The sheets must be joined before lines are drawn across them, or the 
lines will become distorted. Drawing-paper is now made in rolls of great 
length, so as to render this operation unnecessary. 

238. Mounting Maps. A map is sometimes required to be mounted — 
i. e., backed with canvas or muslin. To do this, wet the muslin and stretch 



U2 LAND-SURVEYING. 

it strongly on a board by tacks driven very near together. Cover it with 
strong paste, beating this in with a brush to fill up the pores of the muslin. 
Then spread paste over the back of the paper, and when it has soaked into it 
apply it to the muslin, inclining the board, and pasting first a strip, about 
two inches wide, along the upper side of the paper, pressing it down with 
clean linen in order to drive out all air- bubbles. Press down another strip 
in like manner, and so proceed till all is pasted. Let it dry very gradually 
and thoroughly before cutting the muslin from the board. 

Maps may be varnished with picture-varnish, or by applying four or five 
coats of isinglass-size, letting each dry well before applying the next, and 
giving a full, flowing coat of Canada balsam diluted with the best oil of tur- 
pentine. 

LATITUDES AND DEPARTURES. 

239. Definitions. The Latitude of a point is its distance north 
or sonth of some " Parallel of Latitude" or line running east or 
west. The Longitude of a point is its distance east or west of 
some " Meridian," or line running north and south. In compass- 
surveying, the magnetic meridian — i. e., the direction in which 
the magnetic needle points — is the line from which the longitudes 
of points are measured or reckoned. 

The distance which one end of a line is due north or south of 
the other end is called the Difference of Latitude of the two ends 
of the line; or its northing or southing; or simply its latitude. 

The distance which one end of the line is due east or west of 
the other is here called the Difference of Longitude of the two 
ends of the line ; or its easting or westing ; or its departure. 

Latitudes and Departures are the most usual terms, and will 
be generally used hereafter, for the sake of brevity. 

This subject may be illustrated geographically, by noticing that 
a traveler, in going from New York to Buffalo in a straight line, 
would go about one hundred and fifty miles due north, and two 
hundred and fifty miles due west. These distances would be the 
differences of latitude and of longitude between the two places, or 
his northing and westing. Returning from Buffalo to Xew York, 
the same distances would be his southing and easting.* 



* It should be remembered that the following discussions of the latitudes and lon- 
gitudes of the points of a survey will not be perfectly applicable to those of distant 
places, such as the cities just named, in consequence of the surface of the earth not 
being a plane. 



LATITUDES AND DEPARTURES. 



143 



Fig. 176. 
IS 




In mathematical language,* the operation of finding the lati- 
tude and longitude of a line, from its bearing and length, would 
be called the transformation of Polar Co- 
ordinates into Rectangular Co-ordinates. 
It consists in determining, by our Second 
Principle, the position of a point which 
had originally been determined by the a-— 
Third Principle. Thus, in the figure 

(which is the same as that of Art. 7), the point S is determined 
by the angle SAC and by the distance A S. It is also determined 
by the distances A C and C S, measured at right angles to each 
other ; and then, supposing C S to run due north and south, C S 
will be the latitude, and A C the departure of the line A S. 



\v~ 



240. Calculation of Latitudes and Departures. Let A B be a 

given line, of which the length A B, 
and the bearing (or angle, B A C, 
which it makes with the magnetic me- 
ridian), are known. It is required to 
find the differences of latitude and of 
longitude between its two extremities 
A and B — that is, to find A C and 
p, C B ; or, what is the same thing, B D 
and D A. 

It will be at once seen that A B is 
the hypotenuse of a right-angled tri- 
angle, in which the "Latitude" and 
the "Departure" are the sides about the right angle. We there- 
fore know, from the principles of trigonometry, that 
A C = A B . cos. BAG, 
B C = A B . sin. B A C. 
Hence, to find the latitude of any course, multiply the natural 
cosine of the bearing by the length of the course ; and to find the 
departure of any course, multiply the natural sine of the bearing 
by the length of the course. 

If the course be northerly, the latitude will be north, and will 
be marked with the algebraic sign + , plus, or additive ; if it be 




144 LAND-SURVEYING. 

southerly, the latitude will be south, aud will be marked with the 
algebraic sign — , minus, or subtractiye. 

If the course be easterly, the departure will be east, and 
marked -J-, or additive ; if the course be westerly, the departure 
will be west, and marked — , or subtractive. 

241. Formulas. The rules of the preceding article may be ex- 
pressed thus : 

Latitude = distance X cos. bearing, 
Departure = distance X sin. bearing.* 
From these formulas may be obtained others, by which, when 
any two of the above four things are given, the remaining two 
can be found. 

When the Bearing and Latitude are given; 

Distance = ^^— = latitude X sec. bearing, 

cos. bearing o' 

Departure = latitude X tang, bearing. 
When the Bearing and Departure are given; 

Distance = 8 f n ep b a ^ r D e g = departure X cosec. bearing. 
Latitude = departure X cotang. bearing. 
Wlien the Distance and Latitude are given ; 

r\ i • latitude 

Cos. bearing = — — 

o distance 7 

Departure = latitude X tang, bearing. 
Wlien the Distance and Departure are given; 
Sin. bearing = ^. p f ure , 

o distance ' 

Latitude = departure X cotang. bearing. 
Wlien the Latitude and Departure are given ; 
Tang, of bearing = 5— 
Distance = latitude X sec. bearing. 
Still more simply, any two of these three — distance, latitude, 
and departure — being given, we have 

Distance = ^/ (latitude 2 -f- departure 2 ) 
Latitude = ^/ (distance 2 — departure 2 ) 
Departure = ^/ (distance 2 — latitude 2 ) 

* Whenever sines, cosines, tangents, etc., are here named, they moan the natural 
sines, etc., of an arc described with a radius equal to one, or to the uuit by which the 
sines, etc., are measured. 



LATITUDES AND DEPARTURES. 145 

242. Traverse-Tables. The latitude and departure of any dis- 
tance, for any bearing, could be found by the method given in 
Art. 240, with the aid of a table of natural sines. But to facili- 
tate these calculations, which are of so frequent occurrence and 
of so great use, traverse-tables have been prepared, originally for 
navigators (whence the name traverse), and subsequently for sur- 
veyors.* 

The traverse-table at the end of this volume gives the latitude 
and departure for any bearing, to each quarter of a degree, and for 
distances from 1 to 9. 

To use it, find in it the number of degrees in the bearing, on 
the left-hand side of the page, if it be less than 45°, or on the 
right-hand side if it be more. The numbers on the same line, 
running across the page,f are the latitudes and departures for that 
bearing, and for the respective distances — 1, "2, 3, 4, 5, 6, 7, 8, 9 
— which are at the top and bottom of the page, and which may 
represent chains, links, rods, feet, or any other unit. Thus, if the 
bearing be 15°, and the distance 1, the latitude would be 0*966 
and the departure 0*259. For the same bearing, but a distance of 
8, the latitude would be 7*727 and the departure 2*071. 

Any distance, however great, can have its latitude and depart- 
ure readily obtained from this table ; since, for the same bearing, 
they are directly proportional to the distance, because of the simi- 
lar triangles which they form. Therefore, to find the latitude or 
departure for 60, multiply that for 6 by 10, which merely moves 
the decimal-point one place to the right ; for 500, multiply the 
numbers found in the table for 5, by 100 — i. e., move the decimal- 
point two places to the right, and so on. Merely moving the deci- 
mal-point to the right, one, two, or more places, will therefore 
enable this table to give the latitude and departure for any decimal 
multiple of the numbers in the table. 

* The first traverse-table for surveyors seems to have been published in 1791, by 
John Gale. The most extensive table is that of Captain Boileau, of the British army t 
being calculated for every minute of bearing, and to five decimal places, for distances 
from 1 to 10. The table in this volume was calculated for it, and then compared 
with the one just mentioned. 

f In using this or any similar table, lay a ruler across the page, just above or be- 
low the line to be followed out. This is a very valuable mechanical assistance. 



146 LAND-SURVEYING. 

For compound numbers, such as 873, it is only necessary to 
find separately the latitudes and departures of 800, of 70, and of 
3, and add them together. But this may be done, with scarcely 
any risk of error, by the following simple rule : 

Write down the latitude and departure for the first figure of 
the given number, as found in the table, neglecting the decimal- 
point ; write under them the latitude and departure of the second 
figure, setting them one place farther to the right ; under them 
write the latitude and departure of the third figure, setting them 
one place farther to the right ; and so proceed with all the figures 
of the given number. Add up these latitudes and departures, and 
cut off the three right-hand figures. The remaining figures will 
be the latitude and departure of the given number in links, or 
chains, or feet, or whatever unit it was given in. 

For example : Let the latitude and departure of a course hav- 
ing a distance of 873 links, and a bearing of 20°, be required. In 
the table find 20°, and then take out the latitude and departure 
for 8, 7, and 3, in turn, placing them as above directed, thus : 
Distances. Latitudes. Departures. 

800 7518 2736 

70 6578 2394 

3 2819 1026 

873 820-399 298*566 

Taking the nearest whole numbers and rejecting the decimals, 
we find the desired latitude and departure to be 820 and 299.* 
When a occurs in the given number, the next figure must be 
set two places to the right, the reason of which will appear from the 
following example, in which the is treated like any other number : 
Given a bearing of 35°, and a distance of 3048 links. 
Distances. Latitudes. Departures. 

3000 2457 1721 

000 0000 0000 

40 3277 2294 

__8 6553 458 9 

3048 2496-323 1748-529 

* It is frequently doubtful, in many calculations, when the final decimal is 5, 
whether to increase the preceding figure by one or not. Thus, 43 # 5 may be called 43 
or 44 with equal correctness. It is better, in such cases, not to increase the whole 
number, so as to escape the trouble of changing the original figure, and the increased 



LATITUDES AND DEPARTURES. 147 

Here the latitudes and departures are 2496 and 1749 links. 
, When the bearing is over 45°, the names of the columns must 
be read from the bottom of the page, the latitude of any bearing, 
as 50°, being the departure of the complement of this bearing, or 
40°, and the departure of 40° being the latitude of 50°, etc. The 
reason of this will be at once seen on inspecting Fig. 177, and 
imagining the east and west line to become a meridian. For, if 
A be the magnetic meridian, as before, and therefore B A C be 
the bearing of the course A B, then is A the latitude, and CB 
the departure of that course. But if AE be the meridian and 
BAD (the complement of BAG) be the bearing, then is A D 
(which is equal to C B) the latitude, and D B (which is equal to 
AC) the departure. 

As an example of this, let the bearing be 63 J°, and the distance 
3,469 links. Proceeding as before, we have — 

Distances. Latitudes. Departures. 

3000 1350 2679 

400 1800 3572 

60 2701 5358 

9 4051 8037 



3469- 1561-061 3097'817 

The required latitude and departure are 1561 and 3098 links. 

In the few cases occurring in compass-surveying, in which the 
bearing is recorded as somewhere between the fractions of a degree 
given in the table, its latitude and departure may be found by in- 
terpolation. Thus, if the bearing be lOf °, take the half sum of the 
latitudes and departures for 10J° and 10-£°. If it be 10° 20', add one 
third of the difference between the latitudes and departures for 10J° 
and for 10J°, to those opposite to 10J° ; and so in any similar case. 

The uses of this, table are very varied. The principal applica- 
tions of it, which will now be explained, are to testing the accu- 
racy of surveys ; to supplying omissions in them ; to platting 
them ; and to calculating their content.* 

chance of error. If, however, more than one such case occurs in the same column 
to be added up, the larger and smaller number should be taken alternately. 

* The/ traverse-table admits of many other minor uses. Thus, it may be used 



148 



LAND-SUR VETING. 



243. Application to testing a Survey. It is self-evident that, 
when the surveyor has gone completely around a field or farm, 
taking the bearings and distances of each boundary -line, till he 
has got back to the starting-point, he has gone precisely as far 
south as north, and as far west as east. But the sum of the north 
latitudes tells how far north he has gone, and the sum of the south 
latitudes how far south he has gone. Hence these two sums will 
be equal to each other, if the survey has been correctly made. In 
like manner, the sums of the east and of the west departures 
must also be equal to each other. 

We will apply this principle to testing the accuracy of the sur- 
vey of which Fig. 61 is a plat. Prepare seven columns, and head 
them as below. Find the latitude and departure of each course to 
the nearest link, and write them in their appropriate columns. 
Add up these columns. Then will the difference between the 
sums of the north and south latitudes, and between the sums of 
the east and west departures, indicate the degree of accuracy of 
the survey. 



STATIONS. 


BEARINGS. 


DISTANCES. 


LATITUDES. 


DEPARTURES. 


X. 


s. 


E. W. 


1 

2 
3 
4 
5 


N. 35° E. 
K 83J° E. 
S. 57° E. 
S. 341° W. 

N. oor W. 


2-70 
1-29 
2-22 
3-55 
3-23 

| 


221 
•15 

1-78 


1-21 
2*93 


1-55 I 

1-28 j 

1-86 

2-00 
2-C9 


4-14 | 4-14 


4-69 4-69 



The entire work of the above example is given on the following 
page. 



for solving, approximately, any right-angled triangle by mere inspection, the bearing 
being taken for one of the acute angles ; the latitude being the side adjacent, the de- 
parture the side opposite, and the distance the hypotenuse. Any two of these being 
given, the others are given by the table. The table will therefore serve to show the 
allowance to be made in chaining on slopes (see Art. 20). Look in the column of 
bearings for the slope of the ground — i. e., the angle it makes with the horizon, find 
the given distance, and the latitude corresponding will be the desired horizontal 
measurement, and the difference between it and the distance will be the allowance 
to be made. 



LATITUDES AND DEPARTURES. 149 

34^° 2480 1688 

4133 2814 

4133 2814 



35° 


1638 
57340 ■ 


1147 
40150 


270- 


221-140 


154-850 


83|° 


113 
226 
1019 


994 

1987 
8942 


129- 


14-579 


128-212 


57° 


1089 
1089 
1089 


1677 
1677 
1677 



355- 293-463 199-754 

56i° 1656 2502 

1104 1668 

1656 2502 



323- 178-296 269*382 

The nearest link is taken to be 
inserted in the table, and the re- 
maining* decimals are neglected. 



222- 120-879 186-147 



In the preceding example the respective sums were found to be 
exactly equal. This, however, will rarely occur in an extensive 
survey. If the difference be great, it indicates some mistake, and 
the survey must be repeated with greater care ; but if the differ- 
ence be small it indicates, not absolute errors, but only inaccura- 
cies, unavoidable in surveys with the compass, and the survey may 
be accepted. 

How great a difference in the sums of the columns may be 
allowed, as not necessitating a new survey, is a dubious point. 
Some surveyors would admit a difference of 1 link for every 3 
chains in the sum of the courses ; others only 1 link for every 10 
chains. One writer puts the limit at 5 links for each station ; 
another at 25 links in a survey of 100 acres. But every practical 
surveyor soon learns how near to an equality his instrument and 
his skill will enable him to come in ordinary cases, and can there- 
fore establish a standard for himself, by which he can judge 
whether the difference, in any survey of his own, is probably the 
result of an error, or only of his customary degree of inaccuracy, 
two things to be very carefully distinguished.* 

244. Application to supplying Omissions. Any two omissions 
in the field-notes can be supplied by a proper use of the method of 
latitudes and departures ; as will be explained in Chapter V, which 
treats of "Obstacles to Measurement," under which head this 

* A French writer fixes the allowable difference in chaining at 1-400 of level lines ; 
1-200 of lines on moderate slopes ; 1-100 of lines on steep slopes. 



150 LAXD-SUE YEYIXG. 

subject most appropriately belongs. But a knowledge of the fact 
that any two omissions can be supplied, should not lead the young 
surveyor to be negligent in making every possible measurement, 
since an omission renders it necessary to assume all the notes taken 
to be correct, the means of testing them no longer existing. 

245. Balancing a Survey. The subsequent applications of this 
method require the survey to be previously balanced. This opera- 
tion consists in correcting the latitudes and departures of the 
courses, so that their sums shall be equal, and thus "balance." 
This is usually done by distributing the differences of the sums 
among the courses in proportion to their length : saying, as the 
sum of the lengths of all the courses is to the whole difference of 
the latitude, so is the length of each course to the correction of its 
latitude. A similar proportion corrects the departures.* 

It is not often necessary to make the exact proportion, as the 
correction can usually be made, with sufficient accuracy, by not- 
ing how much per chain it should be, and correcting accord- 
ingly. 

In the example given below, the differences have purposely been 
made considerable. The corrected latitudes and departures have 
been here inserted in four additional columns, but in practice they 
should be written in red ink over the original latitudes and depart- 
ures, and the latter crossed out with red ink. 



to 
Z 

Z BEARINGS. 
t 


X 

■ a 

~ z 


LATITUDES. 


DEPARTURES. 


CORRECTED 
LATITUDES. 


CORRECTED i 
DEPARTURES. 


N.+ S.— 


E.+ W.— 


N.+ ' S. — 


£.+ W.-| 


1 X. 52° E. 

2 S. 29|°E. 

3 S. 31i°AV. 

4 X. 61= W. 


10-63 
4-10 
7*69, 
7-13 


6-54 

i 3-56 
6-54 
3-46: 


8*38 
2-03 

4-05 
6-24 


6-58 

3-55 
6-51 

3-4S 


834 

2-01 

4*08 

6-27 


29-55 


10-00 1010 


10-41 10-29 


10-06 10-06 


10-35 10-35 





The corrections are made by the following proportions : the 
nearest whole numbers being taken : 

* A demonstration of this principle was given by Dr. Bowditeh, in Xo. 4 of " The 

Analyst." 



LATITUDES AND DEPARTURES. 



151 



For 


the Latitudes. 




For the Departures. 


29-55 


: 10-63 


: 10 


4 


29-55 : 10-63 


: 12 : 4 


29-55 


: 4-10 


: 10 


1 


29-55 : 4-10 


: 12 : 2 


29-55 


: 7-69 


: 10 


3 


29-55 : 7-69 


: 12 : 3 


29-55 


: 7-13 


: 10 


2 


29-55.: 7-13 


: 12 : 3 



10 



12 



This rule is not always to be strictly followed. If one line of a 
survey has been measured over very uneven and rough ground, or 
if its bearing has been taken with an indistinct sight, while the 
other lines have been measured over level and clear ground, it is 
probable that most of the error has occurred on that line, and the 
correction should be chiefly made on its latitude and departure. 

If a slight change of the bearing of a long course will favor the 
balancing, it should be so changed, since the compass is much 
more subject to error than the chain. So, too, if shortening any 
doubtful line will favor the balancing, it should be done, since dis- 
tances are generally measured too long. 

246. Application to Platting. Rule three columns ; one for 
stations, the next for total latitudes, and the third for total de- 
partures. Fill the last two columns by beginning at any conven- 
ient station (the extreme east or west is best) and adding up (alge- 
braically) the latitudes of the following stations, noticing that the 
south latitudes are subtractive. Do the same for the departures, 
observing that the westerly ones are also subtractive. 

Taking the example given in Art. 243, and beginning with sta- 
tion 1, the following will be the results : 



STATIONS. 


TOTAL LATITUDES FROM 
STATION 1. 


TOTAL DEPARTURES FROM 
STATION 1. 


1 

2 
3 
4 
5 

1 


o-oo 

+ 2-21 K 
+ 2-36 N. 
+ 1-15 N. 
- 1-78 S. 
0-00 


o-oo 

+ 1-55 E. 
+ 2-83 E. 
+ 4-69 E. 
+ 2-69 E. 

o-oo 



It will be seen that the work proves itself, by the total latitudes 
and departures for station 1, again coming out ' equal to zero. 
To use this table, draw a meridian through the point taken for 



152 



LAND-SUR VEYIXG. 



Fig. 178. 



station 1, as in Fig. 178. Set off, upward from this, along the 
meridian, the latitude, 221 links, to A, and from A, to the right 

perpendicularly, set 
off the departure, 155 
links.* This gives 
the point 2. Join 
1....2. From 1 again, 
set off, upward, 236 
links, to B, and from 
B, to the right, per- 
pendicularly, set off 
283 links, which will 
fix the point 3. Join 
2.... 3 ; and so pro- 
ceed, setting off north 
latitudes along the 
meridian upward, and 
south latitudes along 
it downward ; east 
departures perpendicularly to the right, and west departures per- 
pendicularly to the left. 

The advantages of this method are its rapidity, ease, and accu- 
racy ; the impossibility of any error in platting any one course 
affecting the following points ; and the certainty of the plat "com- 
ing together," if the latitudes and departures have heen "bal- 
anced." 




CALCULATING THE CONTENT. 

247. Methods. When a field has been platted, by "whatever 
method it may have been surveyed, its content can be obtained 
from its plat by dividing it up into triangles, and measuring on the 
plat their bases and perpendiculars ; or by any of the other means 
explained in Chapter II. 

But these are only approximate methods, their degree of accu- 



* This is most easily done with the aid of a right-angled triangle, sliding one of 
the sides adjacent to the right angle along the blade of the square, to which the other 
side will then be perpendicular. 



CALCULATING THE CONTENT. 



153 



racy depending on the largeness of scale of the plat and the skill 
of the draughtsman. The invaluable method of latitudes and de- 
partures gives another means, perfectly accurate, and not requiring 
the previous preparation of a plat. It is sometimes called the 
rectangular, or the Pennsylvania, or Kittenhouse's method of cal- 
culation.* 

248. Definitions. Imagine a meridian line to pass through the 
extreme east or west corner of a field. According to the defi- 
nitions established in Art. 239 (and here recapitulated for con- 
venience of reference), the perpendicular distance of each station 
from that meridian is the Longitude of that station ; additive, or 
plus, if east ; subtractive, or minus, if west. The distance of the 
middle of any line, such as the side 
of the field, from the meridian* is 
called the longitude of that side.f 
The difference of the longitudes 
of the two ends of a line is called 
the Departure of that line. The 
difference of the latitudes of the 
two ends of a line is called the 
Latitude of the line. 



Fig. 179. 




249. Longitudes, To give more 
definiteness to the development of 
this subject, the figure in the mar- 
gin will be referred to, and may be 
considered to represent any space 
inclosed by straight lines. 

Let N S be the meridian passing 
through the extreme westerly station of the field ABODE. 

* It is, however, substantially the same as Mr. Thomas Burgh's " Method to de- 
termine the Areas of Right-lined Figures universally," published nearly a century 
ago. 

f The phrase " meridian distance " is generally used for what is here called " Ion- 
gitude"; but the analogy of "differences of longitude" with "differences of lati- 
tude," usually but anomalously united with the word " departure," borrowed from 
navigation, seems to put beyond all question the propriety of the innovation here 
introduced. 



154 LAHTD-SURVETING. 

From the middle and ends of each side draw perpendiculars to 
the meridian. These perpendiculars will be the longitudes and 
departures of the respective sides. The longitude, FG, of the 
first course, AB, is evidently equal to half its departure, HB. 
The longitude, JK, of the second course, B C, is equal to JL 
+ LM + M K, or equal to the longitude of the preceding course, 
plus half its departure, plus half the departure of the course it- 
self. The longitude, Y Z, of some other course, as E A, taken any- 
where, is equal to WX — YX — TJ V, or equal to the longitude 
of the preceding course, minus half its departure, minus half the 
departure of the course itself — i. e., equal to the algebraic sum of 
these three parts, remembering that westerly departures are nega- 
tive, and therefore to be subtracted when the directions are to 
make an algebraic addition. 

To avoid fractions it will be better to double each of the pre- 
ceding expressions. We shall then have a 

General Eule foe finding DorBLE Longitudes. 

The double longitude of the first course is equal to its depart- 
ure. 

The double longitude of the secoxd course is equal to the double 
longitude of the first course, plus the departure of that course, plus 
the departure of the second course. 

The double longitude of the third course is equal to the double 
longitude of the second course, plus the departure of that course, 
plus the departure of the course itself. 

The double longitude of axt course is equal to the double longi- 
tude of the preceding course, plus the departure of that course, plus 
the departure of the course itself* 

The double longitude of the last course (as well as of the first) 
is equal to its departure. Its " coming out" so, when obtained by 
the above rule, proves the accuracy of the calculation of all the 
preceding double longitudes. 

250. Areas. We will now proceed to find the area or content 
of a field, by means of the "double longitudes" of its sides, which 

* The last course is a " preceding course " to the first course, as will appear on 
remembering that these two courses join each other on the ground. 



CALCULATING TEE CONTENT. 



155 



can be readily obtained by the preceding rule, whatever their 
number. 



Fig. 180. 




251. Beginning with a three-sided field, A B C in the figure, 
draw a meridian through A, and draw perpendiculars to it as in 
the last figure. It is plain that its con- 
tent is equal to the difference of the areas 
of the trapezoid D B C E, and of the tri- 
angles A B D and A C E. 

The area of the triangle A B D is equal 
to the product of AD by half of D B, or 
to the product of AD by E G ; i. e., equal 
to the product of the latitude of the first 
course by its longitude. 

The area of the trapezoid D B C E is 
equal to the product of DE by half the 
sum of DB and CE, or by HJ; i. e., 
to the product of the latitude of the sec- 
ond course by its longitude. 

The area of the triangle A C E is equal to the product of A E 
by half EC, or by K L ; i. e., to the product of the latitude of the 
third course by its longitude. 

Calling the products in which the lati- 
tude was north, North Products, and the 
products in which the latitude was south, 
South Products, we shall find the area of 
the trapezoid to be a south product, and 
the areas of the triangles to be north prod- 
ucts. The difference of the north products 
and the south products is therefore the de- 
sired area of the three-sided field ABC. 

Using the double longitudes (in order to 
avoid fractions) in each of the preceding 
products, their difference will be the double 
area of the triangle ABC. 

252. Taking now a four-sided field, 

A B C D in the figure, and drawing a meridian and longitudes as 
11 



Fig. 181. 




156 LAND-SURVEYING. 

before, it is seen, on inspection, that its area would be obtained 
by taking the two triangles, ABE, ADG, from the figure 
E B C D G E, or from the sum of the two trapezoids E B C F and 
FCDG. 

The area of the triangle A E B will be found, as in the last 
article, to be equal to the product of the latitude of the first 
course by its longitude. The product will be North. 

The area of the trapezoid E B C F will be found to equal .the 
latitude of the second course by its longitude. The product will 
be South. 

The area of the trapezoid FCDG will be found to equal the 
product of the latitude of the third course by its longitude. 
The product will be South. 

The area of the triangle A D G will be found to equal the prod- 
uct of the latitude of the fourth course by its longitude. The 
product will be North. 

The difference of the north and south products will therefore be 
the desired area of the four-sided field ABCD. 

Using the double longitude as before, in each of the preceding 
products, their difference will be double the area of the field. 

253. Whatever the number or directions of the sides of a field, 
or of any space inclosed by straight lines, its area will always be 
equal to half of the difference of the north and south products 
arising from multiplying together the latitude and double longitude 
of each course or side. 

We have, therefore, the following 

General Rule foe finding Aeeas. 

1. Prepare ten columns, headed as in the example below, and in 
the first three write the stations, bearings, and distances. 

2. Find the latitudes and departures of each course, by tlie 
traverse-table, as directed in Art. 242, placing them in the four 
folloioing columns. 

3. Balance them, as in Art. 245, correcting them in red 
ink. 

4. Find the double longitudes, as in Art. 249, with reference 



CALCULATING TEE CONTENT. 



157 



to a meridian passing through the extreme east or west station, and 
place them in the eighth column. 

5. Multiply the double longitude of each course by the corrected 
latitude of that course, placing the north products in the ninth 
column, and the south products in the tenth column. 

6. Add up the last two columns, subtract the smaller sum from 
the larger, and divide the difference by two. The quotient will be 
the content desired. 

254. To find the most easterly or westerly station of a survey, 
without a plat, it is best to make a rough hand-sketch of the sur- 
vey, drawing the lines in an approximation to their true directions, 
by drawing a north and south, and east and west lines, and con- 
sidering the bearings as fractional parts of a right angle, or 90° ; 
a course N. 45° E., for example, being drawn about half-way be- 
tween a north and an east direction ; a course N. 28° W. being not 
quite one third of the way around from north to west ; and so on, 
drawing them of approximately true proportional lengths. 



255. Example 1, given below, refers to the five-sided field, of 
which a plat is given in Fig. 161, and the 
latitudes and departures of which were 
calculated in Art. 243. Station 1 is the 
most westerly station, and the meridian will 
be supposed to pass through it. The double 
longitudes are best found by a continual 
addition and subtraction, as in the mar- 
gin, where they are marked D. L. The 
double longitude of the last course comes 
out equal to its departure, thus proving the 
work. 

The double longitudes being thus ob- 
tained, are multiplied by the corresponding 
latitudes, and the content of the field ob- 
tained as directed in the General Eule. 

This example may serve as a pattern for the most compact 
manner of arranging the work. 



STA- 
TIONS. 


+ 1-55 D.L. 
+ 1-55 

-f 1-28 


1 

2 
3 
4 
5 


+ 438 D. L. 
+ 1-28 
+ 1-86 


+ 7'52 D. L. 
+ 1-86 
-2-00 


+ 7-38 D. L. 
— 2-00 
-2-69 


+ 2-69 D. L. 



158 



LAND-SUR VEYIXG. 



1 * 

E- O 


BEARINGS. 


m 

i H 


LATITUDES. 


DEPARTURES. 


DOUBLE 
LONGI- 
TUDES. 


DOUBLE AREAS. 


N.+ 


s. — 


E.+ 


W.— 


N.— 


S.+ 


1 

2 
3 
4 
5 


N. 35° E. 
N. 83i° e. 
S. 57° E. 
S. 34i° W. 
N. 56i° W. 


2-70 
1-29 
2*22 
3-55 
3-23 


2-21 
•15 

1-78 


1-21 
293 


1-55 
1-28 
1-86 

4-69 


2-00 
2-69 


+ 1-55 

+ 4-38 
+ 7-52 
+ 7-38 
+ 2-69 


3-4255 
0-6570 

4-7882 


9-0992 
21-6234 


4-14 


4-14 


4-69 


8-8707 


30-7226 
8-8707 


Cbnfe»* = 1 A. OR. 15 P. 2)21-8519 














square 


chains, 


10-9259 



STA- 
TIONS. 


— 2-00 D. L. 

— 2-00 

— 2*69 


4 

5 

• 

1 

2 
3 


— 6-69 D. L. 

— 2-69 
+ 1-55 


- 7-83 D. L. 
+ 1-55 
+ 1-28 


— 5-00 D. L. 
+ 1-28 
+ 1-86 


- 1-86 



256. The meridian might equally well 
have been supposed to pass through the 
most easterly station, 4 in the figure. The 
double longitudes could then have been 
calculated as in the margin. They will, of 
course, be all west, or minus. The prod- 
ucts being then calculated, the sum of the 
north products will be found to be 29*9625, 
and of the south products 8-1106, and 
their difference to be 21-8519, the same re- 
sult as before. 

257. A number of examples, with and 
without answers, will now be given as ex- 



ercises for the student, who 
should plat them by some of 
the methods given in the chap- 
ter on platting, using each of 
them at least once. He should 
then calculate their content by 
the method just given, and 
check it, by also calculating the 
area of the plat by some of 
the geometrical or instrument- 
al methods given in Chapter 
I ; for no single calculation is 



Fig. 182. 




CALCULATING TEE CONTENT. 



159 



ever reliable. All the examples (except the last) are from the 
author's actual surveys. 

Example 2, given below, is also fully worked out, as another 
pattern for the student, who need have no difficulty with any 
possible case if he strictly follows the directions which have 
been given. The plat is on a scale of 2 chains to 1 inch 
( = 1:1584). 



i w 

t* o 


BEARINGS. 


92 

• H 
CO V 

S3 


LATITUDES. 


DEPARTURES. 


DOUBLE 
LONGI- 
TUDES. 


DOUBLE AREAS. 


N.+ 


s. — 


E.+ 


w.— 


N.+ 


s.— 


1 

2 
3 
4 
5 
6 


N. 12£° E. 
N. 76° W. 
S. 24^° W. 
S. 48° E. 
S. 12i° E. 
S. 77° E. 

Conti 


2-81 
3-20 
1-14 
1-53 
1-12 
1-64 

mt = 


2-75 
•77 


1-04 
1-02 
1-09 

•37 


•60 

1-1.4 

•24 
1-60 


3-11 
•47 


+ 6-56 
+ 4-05 
+ '47 
+ 1-14 
+ 2-52 
+ 4-36 

c 

chains, 


18-0400 
3-1185 


•4888 
1-1628 
2-7468 
1-6132 


3-52 

A. 3: 


3-52 
B. IP 


3-58 


3-58 
square 


21-1585 

6-0116 

1)15-1469 

7-5734 


6-0116 



Exam fit 



STATIONS. 


BEARINGS. 


DISTANCES. 


1 

2 
3 

4 


N. 52° E. 
S. 29f° E. 
S. 31f° W. 
N. 61° W. 


10-64 
4-09 
7-68 

7-24 



Ans. 4 A. 3R. 28 P. 



Example 4. 



STATIONS. 


BEARINGS. 


DISTANCES. 


1 

2 
3 
4 


S. 21° W. 

N. 83i° E. 
N. 12° E. 

K 47° W. 


12-41 
5-86 
8-25 
4-24 



Ans. 4 A. 2R. 37 P. 



Example 5. 



STATIONS. 


BEARINGS. 


DISTANCES. 


1 

2 
3 
4 
5 


K 34£° E. 
K 85° E. 
S. 56f° E. 
S. 34£° W. 
N. 56i° W. 


2-73 
1-28 
2-20 
3-53 
3-20 



Ans. 1A. OR. 14 P. 





Example 6. 




STATIONS. 


BEARINGS. 


DISTANCES. 


1 

2 
3 
4 
5 
6 


K 35° E. 
S. 56£° E. 
S. 34° W. 
N. 56° W. 
S. 29i° W. 
N. 48i° W. 


6-49 

14-15 

510 

5-84 

2-52 
8-73 



160 



LAND-SUB Y EYING. 



Example 7. 



STATIONS. 


BEARINGS. 


DISTANCES. 


1 


S. 21 i° TV. 


17-62 


2 


S. 34° TV. 


10-00 


3 


N". 56° TV. 


14-15 


4 


K 34° E. 


9-76 


5 


K 67° E. 


2-30 


6 


N". 23° E. 


7-03 


7 


N. 18i° E. 


4-43 


8 


S. 76i°E. 


12-41 





Example 8. 




STATIONS. 


BEARINGS. 


DISTANCES. 


1 


S. 65i° E. 


4-98 


2 


S. 58° E. 


8-56 


3 


S. 14£°TV. 


20-69 


4 


S. 47° W. 


0-60 


5 


S. 57i° TV. 


8-98 


6 


N. 56° TV. 


12-90 


7 


N". 34° E. 


10-00 


8 


X. 21£° E. 


17-62 



Example 9. 



Example 10. 



STATIONS. 


BEARINGS. 


DISTANCES. 


1 


S. 57° E. 


5-77 


2 


S. 36i° TV. 


2-25 


3 


S. 39J° TV. 


1-00 


4 


S. 70i° W. 


1-04 


5 


N. 68|° TV. 


1-23 


6 


ST. 56° TV. 


2-19 


7 


X. 33£° E. 


1-05 


8 


N. oQ±° TV. 


1-54 


9 


X. 33i° E. 


3-18 









TIONS. 


BEARINGS. 


DISTANCES. 


1 


X. 63° 51' TV. 


6-91 


2 


X. 63° 44' TV. 


7-26 


3 


N". 69° 35' TV. 


3-34 


4 


X. 77° 50' TV. 


6-54 


5 


X. 31° 24' E. 


14-38 


6 


X. 31° 18' E. 


16-81 


7 


S. 68° 55' E. 


13-64 


8 


S. 68° 42' E. 


11-54' 


9 


S. 33° 45' TV. 


31-55 



Am. 2 A. K. 32 P. 



Ans. 74 acres. 





Example 11. 




STATIONS. 


BEARINGS. 


DISTANCES. 


1 


X. 18£° E. 


1-93 


2 


X. 9° W. 


1-29 


3 


X. 14° TV. 


2-71 


4 


X. 74° E. 


0-95 


5 


S. 48^° E. 


1-59 


6 


S. 14£° E. 


1-14 


7 


S. 19*° E. 


2-15 


8 


S. 23i° TV. 


1-22 


9 


S. 5° W. 


1-40 


10 


S. 30° W. 


1-02 


11 


S. 81i° TV. 


0-69 


12 


X. 32£° TV. 


1-98 



Example 12. 



1 

STATIONS. 


BEARINGS. 


DISTANCES. 


1 


N. 72f ° E. 


0-88 


2 


S. 20i° E. 


0-22 


3 


S. 63° E. 


0-75 


4 


X. 51° E. 


2-35 


5 


X. 44° E. 


1-10 


6 


X. 25i° TV. 


1-96 


7 


X. U° TV. 


1-05 


8 


S. 29° TV. 


1-63 


9 


X. 7U° TV. 


0-81 


10 


X. 13£° TV. 


117 


11 


X. 63° TV. 


1-28 


12 


West. 


1-68 


13 


X. 49° TV. 


0-80 


14 


s. m E. 


6-20 



Example 13. A farm is described in an old deed as bounded 
thus : Beginning at a pile of stones, and running thence twenty- 



CALCULATING THE CONTENT. 



161 



seven chains and seventy links southeasterly sixty-six and a half 
degrees to a white-oak stump ; thence eleven chains and sixteen 
links northeasterly twenty 

A <L 1* A 4- FlG ' 188 ' 

and a half degrees to a 
hickory-tree ; thence two 
chains and thirty-five links 
northeasterly thirty-six de- 
grees to the southeasterly 
corner of the homestead ; 
thence nineteen chains and 
thirty-two links northeast- 
erly twenty-six degrees to 
a stone set in the ground ; 
thence twenty-eight chains 
and eighty links northwest- 
erly sixty-six degrees to a 
pine-stump ; thence thirty- 
three chains and nineteen 

links southwesterly twenty-two degrees to the place of beginning, 
containing ninety-two acres, be the same more or less. Eequired 
the exact content. 




258. 



Fig. 184 



A*; 



+ A 
+ B 
The 



Mascheroni's Theorem. The surface of any polygon is 
equal to half the sum of the products of its 
sides (omitting any one side) taken two and 
two, into the sines of the angles which those 
sides make with each other. 

Thus, take any. polygon, such as the five- 
sided one in the figure. Express the angle 
which the directions of any two sides, as A B, 
C D, make with each other, thus (A B A D). 
Then will the content of that polygon be, as 
below : 

sin (A B A B C) + A B . D . sin (A B A C D) 
sin (A B A D E) + B C . D . sin (B C A D) 



BC 
DE 



B 
B 
. D E . sin (B C A D E) + D . D E . sin (C D A D E)] 

demonstration consists merely in dividing the polygon into 



162 



LAND-SUE V EYING. 



triangles by lines drawn from any angle (as A) ; then expressing 
the area of each triangle by half the product of its base and the 
perpendicular let fall upon it from the above-named angle ; and 
finally separating the perpendicular into parts which can each be 
expressed by the product of some one side into the sine of the 

angle made by it with another side. 
The sum of these triangles equals the 
polygon. 

The expressions are simplified by 
dividing the proposed polygon into two 
parts by a diagonal, and computing the 
area of each part separately, making 
the diagonal the side omitted. 

A New Method of calculating; 
Areas. 

259. In Fig. 185, let the total lati- 
tudes (Art. 246) of the stations 1, 2, 3, 
and 4 be represented by 1 1} l 2 , 1 3 , and 
Z 4 , respectively. 
Let the departures of each course separately be represented by 
d\> d%, d 3 , and d±, respectively. 
The double area of A B 23 

= A B ( A2 + B3) 
= (h~h) (^1 + ^ + ^2) 
= IA + M + hd-2 - Vk - hdi - kd 2 . [L] 
The double area of C B34 

= CB (B3 + C4) 

= (&+«(*+*+*) 

= hd 4 + l 3 d, + l 3 d 3 + hd, + hd, + hd 3 . [2/J 
The double area of 12 A = Al (A2) = l$ x . [3.] 

The double area of 14 C = Cl (C4) = 7 4 r7 4 . [4.] 

Now, the double area of the figure 1234 is equal to the sum 
of [1] and [2] - the sum of [3] and [4]. 
Combining and reducing, we have : 

Double area of 1234 = h (d t + d 2 ) -f 1 3 (d± + <7 4 + d 3 - d t - d } 
-*)-+«i (* + *). 




CALCULATING TEE CONTENT. 



163 



Noting that d 4: + d 3 = d i + ck, we have, 
Double area of 1234 = I 2 (d t + d 2 ) + l s (d 2 



d 3 ) + k (d 3 + d A ). 



Putting this in the form of a rule, we have : Multiply the total 
latitude of each station by the algebraic sum of the departures of 
the two adjacent courses. One half of the algebraic sum of the 
products will be the area. 

As an exercise for the student, let him find, by the above 
method, an expression for the area of figures having five and six 
sides. 

The following example, worked out by the method of double 
longitudes (on page 158), and below, by the new method, will show 
the difference between the two methods : 



1 °Q 

< * 


BEARINGS. 


CO 

p 4 
EH 


LATITUDES. 


DEPARTURES. 


TOTAL 
LATI- 
TUDES. 


ADJA- 
CENT 
DEPART- 
URES. 


DOUBLE 

AREAS. 


N.+ 


s.— 


E.+ 


w.- 


1 

2 
3 
4 
5 


N. 35° E. 
N. 83£° E. 
S. 57° E. 
S. 34£° W. 
N. 56i° W. 


2-70 
1-29 
2-22 
3-55 
3-23 


2-21 
•15 

1-78 


121 
2-93 


1-55 
1-28 
1-86 


2-00 
2-69 


2-21 
2-36 
1-15 

—1-78 


2-83 

3-14 

—0-14 

—4-69 


6-2543 

7-4104 

-0-1610 

8-3482 


4-14 


4-14 


4-69 


4-69 


2) 21-8519 










square 


chains, 


10-9259 



In computing the total latitudes, if the total latitude of the 
last station equals the latitude of the last course with sign changed, 
the total latitudes may be considered correct. 

The station through which the meridian of the survey is sup- 
posed to pass, and from which the total latitude is reckoned, will 
have no latitude, and hence the product of its latitude and adja- 
cent departures will be zero. There will therefore be one less 
product than there are stations. 

Any station may be taken as the starting-point. 

To verify the area obtained in any case, calculate a second time, 
using a different station as the starting-point. 

This method was first published by J. Woodbridge Davis, 
C. E., Ph. D., in Van Nostrand's "Engineering Magazine," for 
April, 1879, where a general discussion of the method is given. 



164 LAND-SUE VEYIKG. 

THE DECLINATION OF THE MAGNETIC NEEDLE. 

260. Definitions. The magnetic meridian is the direction indi- 
cated by the magnetic needle. The true meridian is a true north 
and south line, which, if produced, would .pass through 
Fig. 186. the poles of the earth. The declination of the needle is 
the angle which one of these lines makes with the other. 
In the figure, if N S represent the direction of the 
true meridian, and W S' the direction of the magnetic 
meridian at any place, then is the angle NAN' the decli- 
nation of the needle at that place. 

261. Direction of the Needle. The directions of these 
g I' two meridians do not generally coincide, but the needle 
in most places points to the east or to the west of the 
true north, more or less according to the locality. Observations 
of the amount and the direction of this declination have been 
made in nearly all parts of the world. In the United States the 
declination in the Eastern States is westerly, and in the "Western 
States is easterly, as will be given in detail, after the methods 
for determining the true meridian, and consequently the declina- 
tions, at any place have been explained. 

To determine the True Meridian. 

262. By Equal Shadows of the Sun. On the south side of any 
level surface erect an upright 
staff, shown in horizontal pro- 
jection at S. Two or three 
hours before noon, mark the 
extremity, A, of its shadow. 
Describe an arc of a circle 
with S, the foot of the staff, for 
center, and SA, the distance 
to the extremity of the shadow, 
for radius. About as many 

hours after noon as it had been before noon when the first mark 
was made, watch for the moment when the end of the shadow 




THE DECLINATION OF TEE MAGNETIC NEEDLE. 165 

touches the arc at another point, B. Bisect the arc A B at N. 
Draw S 1ST, and it will be the true meridian, or north and south 
line required. 

For greater accuracy, describe several arcs beforehand, mark 
the points in which each of them is touched by the shadow, bisect 
each, and adopt the average of all. The shadow will be better de- 
fined if a piece of tin with a hole through it be placed at the top 
of the staff, as a bright spot will thus be substituted for the less 
definite shadow. Nor need the staff be vertical, if from its summit 
a plumb-line be dropped to the ground, and the point which this 
strikes be adopted as the center of the arcs. 

This method is a very good approximation, though perfectly 
correct only at the time of the solstices, about June 21st and De- 
cember 22d. It was employed by the Eomans in laying out cities. 

To get the declination, set the compass at one end of the true 
meridian line thus obtained, sight to the other end of it, and take 
the bearing as of any ordinary line. The number of degrees in the 
reading will be the desired declination of the needle. 

263. By the North Star, when in the Meridian. The north 
star, or pole star (called by astronomers 

T^tp 1 88 

Alpha Ursce Minoris, or Polaris), is not ' ' 

situated precisely at the north pole of the .,- — * — ^ 

heavens. If it were, the meridian could /' 

be at once determined by sighting to it, / \ 

or placing the eye at some distance be- B f ° * 

hind a plumb-line so that this line should 
hide the star. But the north star is about 'v 

\\° from the pole. Twice in twenty-four % 

hours, however (more precisely, twenty- 
three hours fifty-six minutes), it is in the meridian, being then 
exactly above or below the pole, as at A and in the figure. To 
know when it is so, is rendered easy by the aid of another star, 
easily identified, which at these times is almost exactly above or 
below the north star — 1 e., situated in the same vertical plane. 
If, then, we watch for the moment at which a suspended plumb- 
line will cover both these stars, they will then be in the meridian. 



166 



LAFD-SUR VEYING. 



Fig. 189. 



P01B 



Fig. 190. 



+ 






* 



" -¥• 



t 



The other star is in the well-known constellation of the Great 
Bear, called also the Plow, or the Dipper, or Charles's Wain. 

Two of its five bright 
stars (the right-hand ones 
in Fig. 189) are known 
as the " Pointers," from . 
their pointing near to the 
north star, thus assisting 
in finding it. The star 
in the tail or handle, 
nearest to the four which 
form a quadrilateral, is 
the star which comes to 
the meridian at the same 
time with the north star, 
twice in twenty-four hours, as in Fig. 189 or 190. It is known 
as Alioth, or Epsilon Ursce Major is. * 

To determine the meridian by this method, suspend a long 
plumb-line from some elevated point, such as a stick projecting 
from the highest window of a house suitably situated. The plumb- 
bob may pass into a pail of water to lessen its vibrations. South 
of this set up the compass, at such a distance from the plumb-line 
that neither of the stars will be seen above its highest point — i. e., 
in latitudes of 40° or 50°, not quite as far from the plumb-line as 
it is long. Or, instead of a compass, place a board on two stakes, 
so as to form a sort of bench, running east and west, and on it 
place one of the compass-sights, or anything having a small hole in 
it to look through. As the time approaches for the north star to 
be on the meridian (as taken from the table given below) place the 
compass, or the sight, so that, looking through it, the plumb-line 
shall seem to cover or hide the north star. As the star moves one 
way, move the eye and sight the other way, so as to constantly 
keep the star behind the plumb-line. At last Alioth, too, will be 



* The north pole is very nearly at the intersection of the line from Polaris to 
Alioth, and a perpendicular to this line from the small star seen to the left of it in 
Fig. 189. 



THE DECLINATION OF TEE MAGNETIC NEEDLE. 167 

covered by the plumb-line. At that moment the eye and the 
plumb-line are (approximately) in the meridian. Fasten down the 
sight on the board till morning, or with the compass take the bear- 
ing at once, and the reading is the declination. 

Instead of one plumb-line and a sight, two plumb-lines may 
be suspended at the end of a horizontal rod, turning on the top of 
a pole. 

The line thus obtained points to the east of the true line when 
the north star is above Alioth, and vice versa. The north star is 
exactly in the meridian about twenty-five minutes after it has been 
in the same vertical plane with Alioth, and may be sighted to, after 
that interval of time, with perfect accuracy. 

Another bright star, which is on the opposite side of the pole, 
and is known to astronomers as Gamma Cassiopeia, also comes on 
the meridian nearly at the same time as the north star, and will 
thus assist in determining its direction. 

264. The time at which the north star passes the meridian 
above the pole, for every tenth day in the year, is given in the fol- 
lowing table, in common clock-time.* The upper transit is the 
most convenient, since at the other transit Alioth is too high to be 
conveniently observed : 



S 



Months. 



January . . 
February . 
March 

April 

May 

June 

July 

August . . . 
September 
October. . 
November 
December 



1st DAT. 



6 30 p. m, 
4 28 " 
2 37 " 
35 " 

10 33 a.m. 
8 32 " 
6 34 " 
4 33 " 
2 31 " 
34 " 

10 28 p. m, 
8 30 " 



llth Day. 



H. M. 

5 51 p. m 

3 48 

1 58 
11 56 a 

9 54 

7 52 

5 55 

3 53 

1 52 
11 50 p 

9 48 

7 50 



M. 



'i>.. 



21st Day. 



M. 



H. M. 

5 11 P. M 

3 09 < 

1 18 < 
11 16 A. 

9 15 « 

713 ' 

5 15 ' 

3 14 ' 

1 12 ' 

11 11 P.M 

9 09 
711 



* To calculate the time of the north star passing the meridian at its upper cul- 
mination : Find in the " American Ephemeris and Nautical Almanac " the right ascen- 



168 LAND-SURVEYING. 

To find the time of the star's passage of the meridian for other 
days than those given in the table, take from it the time for the 
day most nearly preceding that desired, and subtract from this time 
four minutes for each day from the date of the day in the table to 
that of the desired day ; or, more accurately, interpolate by saying : 
"As the number of days between those given in the table is to 
the number of days from the next preceding day in the table to 
the desired day, so is the difference between the times given in 
the table for the days next preceding and following the desired 
day to the time to be subtracted from that of the next preceding 
day." 

The north star passes the meridian later every year. In 1890 
it will pass the meridian about two minutes later than in 1885 ; in 
1895 six minutes, and in 1900 ten minutes later than in 1885, the 
year for which the preceding table has been calculated. 

The times at which the north star passes the meridian below 
the pole in its lower transit can be found by adding eleven hours 
and fift} T -eight minutes to the time of the upper transit, or by sub- 
tracting that interval from it.* 

265. By the North Star at its Extreme Elongation. When the 
north star is at its greatest apparent angular distance east or west 
of the pole, as at B or D in Fig. 188, it is said to be at its extreme 
eastern or extreme western elongation. If it be observed at either 
of these times, the direction of the meridian can be easily obtained 

sion of the star, and from it (increased by twenty -four hours if necessary to render 
the subtraction possible) subtract the right ascension of the sun at mean noon, or 
the sidereal time at mean noon, for the given day, as found in the " ephemeris of the 
sun " in the same almanac. From the remainder subtract the acceleration of side- 
real on mean time corresponding to this remainder (3m. 56s. for 24 hours), and the 
new remainder is the required mean solar time of the upper passage of the star 
across the meridian, in " astronomical " reckoning, the astronomical day beginning at 
noon of the common civil day of the same date. 

* The north star, which is now about 1° 18' from the pole, was 12 ; distant from 
it when its place was first recorded. Its distance is now diminishing at the rate 
of about a third of a minute in a year, and will continue to do so till it approaches 
to within half a degree, when it will again recede. The brightest star in the 
northern hemisphere, Alpha Lurce, will be the pole-star in about 12,000 years, 
being then within about 5° of the pole, though now more than 51° distant 
from it. 



TEE DECLINATION OF THE MAGNETIC NEEDLE. 169 

from tlie observation. The great advantage of this method over 
the preceding is that then the star's motion apparently ceases for 
a short time. 



MEAN TIME OF THE ELONGATIONS OF POLARIS FOR 18S5, LATITUDE 40° NORTH.* 



DATE. 


EASTERN ELONGATION. 


WESTERN ELONGATION. 


January 1, 1885 

"15, " 


H. M. 

12 35-3 p.m. 
11 36-1 A.M. 


H. M. 

12 24 
11 29 


6 A.M. 
3 P. M. 


February 1, " 

" 15, " 


10 29-0 " 
9 33-7 " 


10 22 

9 27 


2 " 
" 


March 1, " .' 

" 15, " 


8 38-5 " 
7 43'4 " 


8 31 

7 36 


8 " 
6 " 


April 1, " 

" 15, " 


6 36'4 " 
5 41-4 " 


6 29 
5 34 


7 " 
7 " 


May 1, " 

" 15, " 


4 38-6 " 
3 43*7 " 


4 31 
3 36 


8 " 

9 " 


June 1, " 

" 15, " 


2 37'1 " 
1 42-2 " 


2 30 
1 35 


3 kt 

4 " 


July 1, ll 

"15, " 


12 39-6 " 
11 44-7 p.m. 


12 32 
11 34 


8 " 

A. M. 


August 1, " 

" 15, " 


10 38-2 " 
9 43-3 " 


10 27 
9 32 


5 " 

6 " 


September 1, " 

" 15, " 


8 36-6 " 
7 41-7 " 


8 26 
7 31 


lt 

1 " 


October 1, " 

" 15, " 


6 38-9 " 
5 43-9 " 


6 28 
5 33 


2 " 
2 " 


November 1, " 

15, " 


4 37'0 " 
3 41-9 " 


4 26 
3 31 


4 " 
3 " 


December 1, " 

15, " 


2 38-9 " 
1 43-6 " 


2 28 
1 33 


2 " 
" 


January 1, 1886 


12 35-0 " 


12 24 


3 " 



For any other days than those given in the table, interpolate 
directly, or subtract 3 * 94 minutes for every day elapsed. For any 
other year add 0*35 minute for every year. Also add one minute 



* To calculate the times of the greatest elongation of the north star : Find in the 
" American Ephemeris and Nautical Almanac " its polar distance at the given time. 
Add the logarithm of its tangent to the logarithm of the tangent of the latitude of 
the place, and the sum will be the logarithm of the cosine of the hour angle before 
or after the culmination. Reduce the space to time ; correct for sidereal acceleration 
(3m. 56s. for 21 hours) and subtract the result from the time of the star's passing the 
meridian on that day, to get the time of the eastern elongation, or add it to get the 
western. 



170 LAND-SURVEYING. 

if the year is the second after leap-year ; add two minutes if it is 
the third after leap-year ; add three minutes if it is leap-year before 
March 1st, and subtract one minute if it is leap-year after March 
1st. 

For any other latitude than 40° north (between 20° and 
50°) add 0*14 minute for each degree of latitude south of 40°, 
or subtract 0*18 minute for each degree of latitude north of 
40°. 

266. Observations. Knowing from the preceding table the hour 
and minute of the extreme elongation on any day, a little before 
that time suspend a plumb-line, precisely as in Art. 263, and place 
yourself south of it as there directed. As the north star moves 
one way, moye your eye the other, so that the plumb-line shall 
continually seem to cover the star. At last the star will appear 
to stop moving for a time, and then begin to move backward. 
Fix the sight on the board (or the compass, etc.) in the position 
in which it was when the star ceased moving ; for the star was 
then at its extreme apparent elongation, east or west, as the case 
may be. 

The eastern elongations from October to March, and the west- 
ern elongations from April to September, occurring in the daytime, 
they will generally not be visible except with the aid of a powerful 
telescope. 



267. Azimuths. The angle which the line from the eye to the 
plumb-line makes with the true meridian — i. e., the angle between 
the meridian plane and the vertical plane passing through the eye 
and the star — is called the Azimuth of the star. It is given in the 
following table for different latitudes, and for a number of years 
to come. For the intermediate latitudes it can be obtained by 
a simple proportion, similar to that explained in detail in Art. 
264.* 



* To calculate this azimuth : From the logarithm of the sine of the polar distance 
of the star, subtract the logarithm of the cosine of the latitude of the place ; the 
remainder will be the logarithm of the sine of the angle required. The polar distance 
can be obtained as directed in the last note. 



THE DECLINATION' OF THE MAGNETIC NEEDLE. 171 





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12 



172 LAFD-SUR VEYINO. 

268. Setting out a Meridian. When two points in the direction 
of the north star at its extreme elongation have been ob- 

Fig. i9i # ° 

p s' tained, as in Art. 266, the true meridian can be found 
thus : Let A and B be the two points. Multiply the 
natural tangent of the azimuth given in the table by the 
distance A B. The product will be the length of a line 
which is to be set off from B, perpendicular to A B, to 
some point C. A and will then be points in the true 
meridian. This operation may be postponed till morn- 
ing. 

If the directions of both the extreme eastern and ex- 
treme western elongations be set out, the line lying mid- 

A way between them will be the true meridian. 

269. Determining the Declination. The declination would, of 
course, be given by taking the bearing of the meridian thus ob- 
tained, but it can also be determined by taking the bearing of the 
star at the time of the extreme elongation, and applying the fol- 
lowing rules : 

When the azimuth of the star and its magnetic bearing are one 
east and the other west, the sum of the two is the magnetic decli- 
nation, which is of the same name as the azimuth — i. e., east, if that 
be east, and west, if it be west. 

When the azimuth of the star and its magnetic bearing are both 
east or both west, their difference is the declina- 
tion, which will be of the same name as the azi- 
muth and bearing, if the azimuth be the greater 
of the two, or of the contrary name if the azi- 
muth be the smaller. 

All these cases are presented together in the 
figure, in which P is the north pole, Z the place 
of the observer, Z P the true meridian, S the star 
at its greatest eastern elongation, and Z X, Z X , 
Z W various supposed directions of the needle. 

Call the azimuth of the star — i. e., the angle 
PZS— 2° east. 

Suppose the needle to point to X, and the 




THE DECLINATION OF THE MAGNETIC NEEDLE. 173 

bearing of the star — i. e., SZN" — to be 5° west of magnetic north. 
The declination PZN will evidently be 7° east of true north. 

Suppose the needle to point to JSP, and the bearing of the star 
— i. e., N'ZS — to be 1£° east of magnetic north. The declination 
will be f ° east of true north, and of the same name as the azimuth, 
because that is greater than the bearing. 

Suppose the needle to point to W, and the bearing of the star 
— i. e., W Ti S — to be 10° east of magnetic north. The declination 
will be 8° west of true north, of the contrary name to the azimuth, 
because that is the smaller of the two.* 

If the star were on the other side of the pole, the rules would 
apply likewise. 

270. Other Methods. Many other methods of determining the 
true meridian are employed ; such as by equal altitudes and azi- 
muths of the sun, or of a star ; by one azimuth, knowing the 
time ; by observations of circumpolar stars at equal times before 
and after their culmination, or before and after their greatest elon- 
gation, etc. 

All these methods, however, require some degree of astronomi- 
cal knowledge ; and those which have been explained are abun- 
dantly sufficient for all the purposes of the ordinary land-surveyor. 

" Burt's Solar Compass" is an instrument by which, "when 
adjusted for the sun's declination and the latitude of the place, the 
azimuth of any line from the true north and south can be read off, 
and the difference between it and the bearing by the compass will 
then be the variation." 

271. Magnetic Declination in the United States. The declina- 
tion in any part of the United States can be approximately ob- 
tained by mere inspection of the map at the beginning of this 
volume, f Through all the places at which the needle, in 1885, 
pointed to the true north, a line is drawn on the map, and called 

* Algebraically, always subtract the bearing from the azimuth, and give the re- 
mainder its proper resulting algebraic sign. It will be the declination ; east if plus, 
and west if minus. Thus, in the first case above, the declination — + 2° — (— 5°) 
= 4- 7° =7° east. In the second case, the declination = + 2° — ( + 1J°) = + f ° = £° 
east. In the third case, the declination = + 2° — ( + 10°) = — 8° = 8° west. 

f Copied from " United States Coast and Geodetic Survey Report," 1882. 



174 LAND-SURVEYING. 

the line of no declination. It will be seen to pass a little east of 
Charleston, South Carolina, thence in a northwesterly direction, 
passing near Zanesville, Ohio, through the west end of Lake Erie, 
passing. a little west of Detroit, and up through the east end of 
Lake Superior. This line is now slowly moving westward. 

At all places situated to the east of this line (including the 
New England States, New York, New Jersey, Delaware, Mary- 
land, Pennsylvania, most of Virginia, and the east half of North 
Carolina and Ohio) the declination is westerly — i. e., the north end 
of the needle points to the west of the true north. At all places 
situated to the west of this line (including the Western and South- 
ern States) the declination is easterly — i. e., the north end of the 
needle points to the east of the true north. This declination in- 
creases in proportion to the distance of the place on either sidcof 
the line of no variation, reaching 23° of easterly declination in 
Washington Territory, and 21° of westerly declination in Maine. 

Isogonics, or lines of equal declination, are lines drawn through 
all the places which have the same declination. On the map they 
are drawn for each degree. All the places situated on the line 
marked 5°, east or west, have 5° declination ; those on the 10° line 
have 10° declination, etc. The declination at the intermediate 
places can be approximately estimated by the eye. These lines all 
refer to 1885. 

The sign -(- indicates west declination, and the sign — indi- 
cates east declination. The annual change in the secular variation 
for stations is given in minutes and decimals, a + indicating in- 
creasing west declination or decreasing east declination, and a — 
sign indicating increasing east and decreasing west declination. 

272. To correct Magnetic Bearings. The declination at any 
place and time being known, the magnetic bearings taken there 
and then may be reduced to their true bearings by these rules : 

Rule 1. When the declination is west, as it is in the North- 
eastern States, the true bearing will be the sum of the declination, 
and a bearing which is north and west, or south and east : and the 
difference of the declination and a bearing which is north and 
east, or south and west. To apply this to the cardinal points, a 



THE DECLINATION OF THE MAGNETIC NEEDLE. 175 



bearing must be called N. O c 



Fig. 193. 

N 

A 






W- 



w- 



north 

west, an east bearing N. 90° E., a 
south bearing S. 0° E., and a west 
bearing S. 90° W. ; counting around 
from W to N", in the figure, and so 
onward, "with the sun." 

The reasons for these corrections 
are apparent from the figure, in 
which the dotted lines and the ac- 
cented letters represent the direction 

of the needle, and the full lines and the unaccented letters repre- 
sent the true north and south and east and west lines. 

When the sum of the declination and the bearing is directed to 
be taken, and comes to more than 90°, the supplement of the sum 
is to be taken, and the first letter changed. When the difference 
is directed to be taken, and the declination is greater than the 
bearing, the last letter must be changed. A diagram of the case 
will remove all doubts. Examples of all these cases are given be- 
low for a declination of 8° west: 






MAGNETIC 


TRUE 


MAGNETIC 


TRTJK 


BEARINGS. 


BEARINGS. 


BEARINGS. 


BEARINGS. 


North. 


N. 8° W. 


South. 


S. 8° E. 


N". 1°E. 


K 7° W. 


S. 2° W. 


S. 6° E. 


N. 40° E. 


N. 32° E. 


S. 60° W. 


S. 52° W. 


East. 


N. 82° E. 


West. 


S. 82° W. 


S. 50° E. 


S. 58° E. 


K 70° W. 


N. 78° W. 


S. 89° E. 


N. 83° E. 


N. 83° W. 


S. 89° W. 



Fig. 194. 

N 



M 



#' 



Eule 2. When the declination is 
east, as in the Western and Southern 
States, the preceding directions must 
be exactly reversed — i. e., the true 
bearing will be the difference of the 
declination, and a bearing which is 
north and west or south and east ; 
and the sum of the declination and a 
bearing which is north and east or 
south and west. A north bearing 



176 



LAND-SUE VEYING. 



must be called N. 0° E., a west bearing N. 90° W., a south bear- 
ing S. 0° W., and an east bearing S. 90° E., counting from N' 
to N, and so onward, "against the sun." The reasons for these 
rules are seen in the figure. Examples are given below for a 
declination of 5° E. : 



MAGNETIC 


TRUE 


MAGNETIC 


TRUE 


BEARINGS. 


BEARINGS. 


BEARINGS. 


BEARINGS. 


North. ' 


N. 5°E. 


South. 


S. 5° W. 


N. 40° E. 


N. 45° E. 


S. 60° W. 


S. 65° W. 


N. 80° E. 


S. 86° E. 


S. 87° W. 


K 88° W. 


East. 


S. 85° E. 


West. 


N. 85° W. 


S. 1° E. 


S. 4° W. 


K 70° W. 


N. 65° W. 


S. 50° E. 


S. 45° E. 


N. 2° W. 


N. 3°E. 



273. To survey a Line with True Bearings. The compass 
may be set, or adjusted, by means of the vernier, according to the 
declination in any place, so that the bearings of any lines then 
taken with it will be their true bearings. To effect this, turn 
aside the compass-plate by means of the tangent-screw which 
moves the vernier a number of degrees equal to the declination, 
moving the south end of the compass-box to the right (the north 
end being supposed to go ahead) if the declination be westerly, 
and vice versa; for. that moves the north end of the compass-box 
in the contrary direction, and thus makes a line which before was 
N. by the needle, now read, as it should truly, north, so many 
degrees west if the declination was west ; and similarly in the 
reverse case. 



Variations of Magnetic Declination. 

274. The variations of the declination are of more practical 
importance than its absolute amount. They are of four kinds : 
Irregular, diurnal, annual, and secular. 



275. Irregular Variation. The needle is subject to sudden 
and violent changes, which have no known law. They are some- 
times coincident with a thunder-storm, or an aurora borealis 



TEE DECLINATION OF TEE MAGNETIC NEEDLE. 177 

(during which changes of nearly 1° in one minute, 2|-° in eight 
minutes, and 10° in one night, have been observed), but often 
have no apparent cause, except an otherwise invisible " mag- 
netic storm." 

276. The Diurnal Variation. On continuing observations of 
the direction of the needle throughout an entire day, it will be 
found, in the northern hemisphere, that the north end of the 
needle moves westward from about 8 a. m. till about 1|- p. m., 
over an arc of from 5' to 15', and then gradually returns to its 
former position. A similar but smaller movement takes place 
during the night. At Philadelphia, the most easterly deflection 
of the needle is at about 7f a. m. The north end of the needle 
then begins to move toward the west, crossing the mean mag- 
netic meridian about 10 J A. m., and reaching its extreme west- 
ern position about 1J- P. M. The total angular range averages 
about 8', being 10J' in August, and 6' in November.* The 
period of this change being a day, it is called the Diurnal Vari- 
ation. Its effect on the permanent variation is necessarily to 
cause it, in places where it is west, to attain its maximum at 
about 1J P. m., and its minimum at about 8 A. m. ; and the 
reverse where the declination is east. 

This diurnal variation adds a new element to the inaccuracies 
of the compass, since the bearings of any line taken on the same 
day, at a few hours' interval, might vary a quarter of a degree, 
which would cause a deviation of the end of the line, amounting to 
nearly half a link at the end of a chain, and to 35 links, or 23 feet, 
at the end of a mile. The hour of the day at which any important 
bearing is taken should therefore be noted. 

277. The Annual Variation. If the observations be continued 
throughout an entire year, it will be found that the diurnal changes 
vary with the seasons, being greater in summer than in winter. 
The period of this variation being a year, it is called the Annual 
Variation. 

* For table of hourly variation of the declination, see " Report of United States 
Coast and Geodetic Survey," 1881, p. 136. 



178 LAND-SUE VEYING. 

278. The Secular Variation. When accurate observations 
on the declination of the needle in the same place are con- 
tinued for several years, it is found that there is a continual 
and tolerably regular increase or decrease of the declination, 
continuing to proceed in the same direction for so long a 
period, that it may be called the Secular Variation of the decli- 
nation. 

The most ancient observations are those taken in Paris. In the 
year 1541 the needle pointed 7° east of north ; in 1580 the declina- 
tion had increased to 11^° east, being its maximum ; the needle 
then began to move westward, and in 1666 it had returned to the 
meridian ; the declination then became west, and continued to in- 
crease till in 1814 it attained its maximum, being 22° 34' west of 
north. It is now decreasing, and, January 1, 1879, it was 16° 56' 
west. 

In this country the north end of the needle was moving east- 
ward at the earliest recorded observations, and continued to do so 
till about the year 1810 (variously recorded as from 1765 to 1819), 
when it began to move westward, which it has ever since con- 
tinued to do. Thus, in Boston, from 1700 to 1807, the declination 
changed from 10° west to 6° 5' west, and, from 1807 to 1879, it 
changed from 6° 5' west to 11° .36' west. 

In Philadelphia, from 1701 to 1802, the declination changed 
from 8° 30' west to 1° 30' west, and, from 1S02 to 1877, it changed 
from 1° 30' west to 6° 2' west. 

Extensive tables of the declination, at more than two thousand 
stations, in various parts of the United States, are given in the 
" Keport of the United States Coast and Geodetic Survey." 1882, 
Appendix XIII, by Charles A. Schott. The secular variation is 
noted on the declination-map in this volume. 

An examination of the above-mentioned tables will show that 
the secular variation often differs greatly in places not far apart. 
and that it varies in amount at the same place from year to 
year: 



THE DECLINATION OF THE MAGNETIC NEEDLE. 179 



TABLE OF COMPUTED ANNUAL CHANGES IN DECLINATION. 



LOCALITY. 



Portland, Me 

Burlington, Vt 

Portsmouth, K H 

Boston, Mass 

Hartford, Conn 

Albany, N. Y 

New York, N. Y 

Buffalo, N. Y 

Philadelphia, Pa 

Baltimore, Md 

Washington, D. C 

Cleveland, Ohio 

Detroit, Mich 

St. Louis, Mo 

Cape Henry, Va 

Charleston, S. C 

Savannah, Ga 

Key West, Fla 

Mobile, Ala 

New Orleans, La 

San Francisco, Cal ' 

Cape Disappointment, W. Ter 
Sitka, Alaska 



ANNUAL CHANGE. 



1870. 



+ 2-4' 
+ 5-0 
+ 4-4 
+ 3-4 
+ 3-8 
+ 4-3 
4 
1 
9 
9 
5 
8 
4 
4 
«s 
5 
6 
3 



+ 2 
+ 5 
+ 4 
+ 3 
+ 3 
+ 2 
+ 3 
+ 3 
+ 3 
+ 3 
+ 3 
+ 4 
+ 2-8 



+ 3 
i 

— 3 

+ 1 



1880. 



+ 1-6' 
+ 6-0 



+ 3 
+ 2 
+ 3 
+ 3 
+ 2 
+ 5-0 
+ 4-9 
+ 3-6 
+ 3 
+ 2 
+ 3 
+ 3 
+ 3 
+ 3'0 
+ 3-5 
+ 4-2 
+ 3 
+ 3 
— 



+ 2 



1885. 



+ 1 

+ 5 
+ 3 
+ 2 
+ 3 
+ 3 
+ 2 
+ 4 
+ 5 
+ 3 
+ 3 
+ 2 
+ 2 
+ 3 
+ 3 
+ 2 
+ 3 
+ 4 
+ 3 
+ 3 
— 
—2 
+ 2 



279. Determination of the Change, by Interpolation, To de- 
termine the change at any place and for any interval not found in 
the recorded observations, an approximation, sufficient for most 
purposes of the surveyor, may be obtained by interpolation (by a 
simple proportion) between the places given on the map, assuming 
the movements to have been uniform between the given dates, and 
also assuming the change at any place not found on the map to 
have been intermediate between those of the lines of equal varia- 
tion, which pass through the places of recorded observations on 
each side of it, and to have been in the ratio of its respective dis- 
tances from those two lines ; for example, taking their arithmetical 
mean, if the required place is midway between them ; if it be twice 
as near one as the other, dividing the sum of twice the change of 
the nearest line, and once the change of the other, by three ; 
and so in other cases — i. e., giving the change at each place, a 



180 LAND-SURVEYIXG. 

"weight" inversely as its distance from the place at which the 
change is to be found. 

280. Determination of the Change by Old Lines. When the 
former bearing of any old line, such as a farm-fence, etc., is re- 
corded, the change in the declination from the date of the original 
observation to the present time can be at once found by setting the 
compass at one end of the line and sighting to the other. The 
difference of the two bearings is the required change. 

If one end of the old line can not be seen from the other, as is 
often the case when the line is fixed only by a " corner " at each' 
end of it, proceed thus : Kun a line from one corner with the old 
bearing and with its distance. Measure the distance from -the end 
of this line to the other corner, to which it will be opposite. Mul- 
tiply this distance by 57 '3, and divide by the length of the line. 
The quotient will be the change of variation in degrees.* 

For example, a line 63 chains long, in 1827 had a bearing of 
north 1° east. In 1847 a trial line was run from one end of the 
former line with the same bearing and distance, and its other end 
was found to be 125 links to the west of the true corner. The 

1 *25 X 57 '3 

change of declination was therefore — — — = 1*137° = 1° 8' 

oo 

westerly. 

281. Effects of the Secular Change. These are exceedingly im- 
portant in the resurvey of farms by the bearings recorded in old 
deeds. Let S N" denote the direction of the needle at the time of 
the original survey, and S' W its direction at the time of the re- 
survey, a number of years later. Suppose the change to have been 

* Let A B be the original line ; A C the trial 

Fig. 195. line, and B C the distance between their extremities. 

A.B and AC may be regarded as radii of a circle 

and B C as a chord of the arc which subtends their 

angle. Assuming the chord and arc to coincide 

(which they will, nearly, for small angles), we have 

' this proportion : Whole circumference : are B C : : 

360° : B A C : or, 2 x A C x 31416 : B C : 360° : 

B C 
BAC, whence B AC =—5 x 5*73 ; or, more precisely, 57 - 29578. 
.a. a 



THE DECLINATION OF TEE MAGNETIC NEEDLE. 181 



Fig. 196. 



^> 



\ 






1* 



s' 



3°, the needle pointing so much 

farther to the west of north. 

The line SN, which before was 

due north and south by the nee- 
dle, will now bear N. 3° E. and 

S. 3° W. ; the line A B, which 

before was N. 40° E. will now 

bear N. 43° E. ; the line D F, 

which before was N". 40° W., will 

now bear N. 37° W., and the line 

WE, which before was due east 

and west, will now bear S. 87° E. 

and N. 87° W. Any line is sim- 
ilarly changed. The proof of this is apparent on inspecting the 

figure . 

Suppose, now, that a surveyor, ignorant or neglectful of this 

change, should attempt to run out a farm by the old bearings of 

the deed, none of the old fences or corners remaining. The full 

lines in the figure represent the 
original bounds of the farm, and 
the dotted lines those of the new 
piece of land which, starting from 
A, he would unwittingly run out. 
It would be of the same size and 
the same shape as the true one, 
but it would be in the wrong place. 
None of its lines would agree with 
the true ones, and in some places 
it would encroach on one neighbor, 
and in other places would leave a 
gore, which belongs to it, between it- 
self and another neighbor. Yet this 
is often done, and is the source of a 

great part of the litigation among farmers respecting their " lines." 

282. To run out Old Lines. To succeed in retracing old lines, 
proper allowance must be made for the change in the variation 




182 LAND-SURVEYING. 

since the date of the original survey. That date must first be ac- 
curately ascertained ; for the survey may be much older than the 
deed, into which its bearings may have been copied from an older 
one. The amount and direction of the change is then to be ascer- 
tained by the methods of Art. 279 or 280. The bearings may then 
be corrected by the following Eules : 

When the north end of the needle has been moving westerly, 
the present bearings will be the sums of the change and the old 
bearings which were northeasterly or southwesterly, and the differ- 
ences of the change and the old bearings which were northwesterly 
or southeasterly. 

If the change has been easterly, reverse the preceding rules, 
subtracting where it is directed to add, and adding where it is 
directed to subtract. 

Kim out the lines with the bearings thus corrected. 

It will be noticed that the process is precisely the reverse of 
that in Art. 272. The rules, there given in more detail, may there- 
fore be used : Eule 1. "When the declination is west," being em- 
ployed when the change has been a movement of the N. end of the 
needle to the east ; and Eule 2, "when the declination is east," 
being employed when the N. end of the needle has been moving to 
the west. 

If the compass has a vernier, it can be set for the change, once 
for all, precisely as directed in Art. 273, and then the courses can 
be run out as given in the deed, the correction being made by the 
instrument. 

Example. The following is a remarkable case which came be- 
fore the Supreme Court of New York : The north line of a large 
estate was fixed by a royal grant, dated in 1704, as a due east and 
west line. It was run out in 1715, by a surveyor, whom we will 
call Mr. A. It was again surveyed in 1765, by Mr. B., who ran a 
course N. 87° 30' E. It was run out for a third time in 1789, by 
Mr. C, who adopted the course X. 86° 18' E, In 1815 it was sur- 
veyed for the fourth time by Mr. D., with a course of X. 88° 30' E. . 
He found old " corners," and " blazes " of a former survey, on his 
line. Thev are also found on another line, south of his. Which 



THE DECLINATION OF THE MAGNETIC NEEDLE. 183 

of the preceding courses were correct, and where does the true 
line lie ? 

The question was investigated as follows : There were no old 
records of variation at the precise locality, but it lies between the 
lines of equal variation which pass through New York and Boston, 
its distance from the Boston line being about twice its distance 
from the New York line. The records of those two cities (re- 
ferred to in Art. 278) could therefore be used in the manner ex- 
plained in Art. 279. For the later dates, observations at New 
Haven could serve as a check. Combining all these, the author 
inferred the variation at the desired place to have been as fol- 
lows : 

In 1715, variation 8° 02' west. 

In 1765, " 5° 32' " Decrease since 1715, 2° 30'. 

In 1789, " 5° 05' " Decrease since 1765, 0° 27'. 

In 1845, . " 7° 23' " Increase since 1789, 2° 18'. 

We are now prepared to examine the correctness of the allowances 
made by the old surveyors. 

The course run by Mr. B. in 1765, N. 87° 30' E., made an 
allowance of 2° 30' as the decrease of variation, agreeing precisely 
with our calculation. The course of Mr. 0. in 1789, N. 86° 18' E., 
allowed a change of 1° 12', which was wrong by our calculation, 
which gives only about 27', and was deduced from three different 
records. Mr. D., in 1845, ran a course of N. 88° 30' E., calling 
the increase of variation since 1789, 2° 12'. Our estimate was 2° 
18', the difference being comparatively small. Our conclusion, 
then, is this : The second surveyor retraced correctly the line of 
the first ; the third surveyor ran out a new and incorrect line ; 
and the fourth surveyor correctly retraced the line of the third, 
and found his marks, but this line was wrong originally, and 
therefore wrong now. All the surveyors ran their lines on the 
supposition that the original "due east and west line" meant 
east and west as the needle pointed at the time of the original 
survey. 

The preponderance of the testimony as to old landmarks agreed 
with the results of the above reasoning, and the decision of the 
court was in accordance therewith. 



184 LAND-SURVEYING. 



I 




jj In the figure below, the horizontal and vertical lines 

represent true east and north lines ; and the two upper lines 

running from left to right represent the two lines set out by 

Fig. 198. __ the survey- 

Xsn^v^ 65 tne years 

there named. 



283. Remedy for the Evils of the Secular Change. The only 
complete remedy for the disputes, and the uncertainty of bounds, 
resulting from the continued change in the declination, is this : 
Let a meridian — i. e., a true north and south line — be established 
in every town or county, by the authority of the State ; monu- 
ments, such as stones, set deep in the ground, being placed at each 
end of it. Let every surveyor be obliged by law to test his com- 
pass by this line, at least once in each year, at a given hour in the 
day. This he could do as easily as in taking the bearing of a fence, 
by setting his instrument on one monument, and sighting to a staff 
held on the other. Let the variation thus ascertained be inserted 
in the notes of the survey, and recorded in the deed. Another 
surveyor, years or centuries afterward, could test his compass by 
taking the bearing of the same monuments, and the difference be- 
tween this and the former bearing would be the change of decli- 
nation. He could thus determine with entire certainty the proper 
allowance to be made (as in Art. 282) in order to retrace the origi- 
nal line, no matter how much, or how irregularly, the declination 
may have changed, or how badly adjusted was the compass of the 
original survey. Any permanent line employed in the same man- 
ner as the meridian line would answer the same purpose, though 
less conveniently, and every surveyor should have such a line, at 
least for his own use.* 



* This remedy seems to have been first suggested by Rittenhouse. It has since 
been recommended by T. Sopwith, in 1822 ; by E. F. Johnson, in 1SS1, and byW. 
Roberts, of Tioy, in 1839. The errors of resurveys, in which the change is neglected; 
were noticed in the " Philosophical Transactions," as long ago as 1679. On magnetic 
declination, see the following " Reports of the United States Coast and Geodetic Sur- 



CHAPTER IV. 

TRANSIT-SURVEYING — BT THE THIRD METHOD. 

THE INSTRUMENTS. 

284. The Tkansit is a Goniometer, or Angle-Measurer. It 
consists, essentially, of a circular plate of metal, supported in such 
a manner as to be horizontal, and divided on its outer circumfer- 
ence into degrees and parts of degrees. Through the center of 
this plate passes an upright axis, and on it is fixed a second circu- 
lar plate, which nearly touches the first plate, and can turn freely 
around to the right and to the left. This second plate carries a 
telescope, which rests on upright standards firmly fixed to the 
plate, and which can be pointed upward and downward. By the 
combination of this motion and that of the second plate around 
its axis, the telescope can be directed to any object. The second 
plate has some mark on its edge, such as an arrow-head, which 
serves as a pointer or index for the divided circle, like the hand of 
a clock. When the telescope is directed to one object, and then 
turned to the right or to the left, to some other object, this index 
which moves with it, and passes around the divided edge of the 
other plate, points out the arc passed over by this change of direc- 
tion, and thus measures the horizontal angle made by the lines 
imagined to pass from the center of the instrument to the two 
objects. 

The great value of this instrument, and the accuracy of its 
measurements of angles, are due chiefly to two things : to the tele- 
scope with its cross-hairs, by which great precision in sighting to a 
point is obtained ; and to the vernier scale, which enables minute 
portions of any arc to be read with ease and correctness. The 



186 



LAND-SUR VEY1NG. 



former assists the eye in directing the line of sight, and the latter 
aids it in reading off the results. Arrangements for giving slow 



Fig. 199. 




and steady motion to the movable parts of the instrument add to 
the value of the above. A contrivance for repeating the observa- 



TEE INSTRUMENTS. 187 

tion of angles still further lessens the unavoidable inaccuracies of 
these observations. 

285. The Surveyor's Transit (Fig. 199). In this instrument 
the telescope takes the place of the plain sights of the surveyor's 
compass, and the angles are read on the graduated limb to single 
minutes by the vernier. 

A level is attached to the telescope, and a vertical circle is 
attached to the telescope-axis inside of the left-hand standard. 
The vertical angles through which the telescope is moved may be 
read off from the vernier attached to the left-hand standard, and 
shown below the vertical circle. The slow-motion screw for the 
vertical circle is shown attached to the right-hand standard. The 
clamp for the axis is hidden by the telescope. The standards upon 
which the telescope-axis rests are fastened to the upper plate (the 
vernier-plate). This plate also carries the compass-circle. The 
compass-circle with its accessories is similar to that already ex- 
plained in the Surveyor's Compass. The compass-circle can be 
turned on its center, so that the declination of the needle can be 
set off, and lines can be run with their true bearings. The vernier- 
plate covers the lower plate (the divided limb), so that only two 
short arcs of the divided limb are seen through openings where the 
verniers are placed. The screw which clamps the vernier-plate to 
the divided limb is shown on the right of the plate, together with 
the slow-motion screw. The lower clamp and the slow-motion 
screw are attached to the upper parallel plate. 

286. As the value of this instrument depends greatly on the 
accurate fitting and bearings of the two concentric vertical axes, 
and as their connection ought to be thoroughly understood, a 
vertical section through the body of the instrument is given in 
Eig. 200. 

The upper plate, or vernier-plate, A, A, carries the verniers, 

compass-box, and telescope. It is attached to its socket by the 

flange, K. This socket is fitted to the outside, conical surface 

of the main socket, C. The main socket, to which is attached 

the divided limb, B, B, is fitted to the conical spindle H, and held 
13 



188 



LAND-SUB VEYING. 



on the spindle by the spring-catch S. A screw holds the coni- 
cal center, whose upper flange keeps the sockets of the two plates 



Fig. 200. 




together. The clamp is at F. Two of the four leveling screws 
are shown in section. The spindle, H, passes through the upper 
parallel plate, and is attached to a movable section of the lower 
parallel plate by a ball-and-socket joint. The leveling screws 
pass through the upper parallel plate, and. rest in cups on the 
lower parallel plate. As the leveling screws are movable on the 
lower parallel plate, the movable section of this plate enables 
the upper part of the instrument to be moved from side to side, 
so as to bring the center of the instrument precisely over any 
desired point. This arrangement is called a " shifting center." 
At the lower end of the spindle is a loop, P, from which the 
plumb-bob is suspended. 

287. The Telescope. This is a combination of lenses, placed in 
a tube, and so arranged, in accordance with the laws of optical 
science, that an image of any object to which the telescope may be 
directed, is formed within the tube (by the rays of light coming 
from the object and bent in passing through the object-glass), and 



TEE INSTRUMENTS. 



189 



there magnified by an eye-glass, or eye-piece, composed of several 
lenses. The arrangement of these lenses is very vari- 

Fig 201 

ous. Those two combinations, which are preferred 
for surveying instruments, will be here explained : 

Fig. 201 represents a telescope which inverts ob- 
jects. Any object is rendered visible by every point 
of it sending forth rays of light in every direction. 
In this figure the highest and lowest points of the 
object, which here is an arrow, A, are alone consid- 
ered. Those of the rays proceeding from them, which 
meet the object-glass, 0, form a cone. The center 
line of each cone, and its extreme upper and lower 
lines, are alone shown in the figure. It will be seen 
that these rays, after passing through the object-glass, 
are refracted or bent by it, so as to cross one another, 
and thus to form at B an inverted image of the object. 
This would be rendered visible, if a piece of ground- 
glass, or other semi-transparent substance, were placed 
at the point B, which is called the focus of the object- 
glass. The rays which form this image continue on- 
ward and pass through the two lenses C and D, which 
act like one magnifying-glass, so that the rays, after 
being refracted by them, enter the eye at such angles 
as to form there a magnified and inverted image of 
the object. This combination of the two plano-convex- 
lenses, C and D, is known as " Ramsden's Eye-piece." 

This telescope, inverting objects, shows them up- 
side down, and the right side on the left. They can 
be shown erect by adding one or two more lenses, as 
in the marginal figure. But as these lenses absorb 
light and lessen the distinctness of vision, the former 
arrangement is sometimes preferred. A little practice 
makes it equally convenient for the observer, who 
soon becomes accustomed to seeing his fiasrtnen stand- 
ing on their heads, and soon learns to motion them 
to the right when he wishes them to go to the left, 
and vice versa. 




190 



LAND-SUE VEYING. 



Fig. 202. Fig. 202 represents a telescope which shows objects 

erect. Its eye-piece has four lenses. The eye-piece of 
the common terrestrial telescope, or spy-glass, has 
three. Many other combinations may be used, all 
intended to show the object achromatically, or free 
from false coloring, but the one here shown is that 
most generally preferred at the present da}\ It will 
be seen that an inverted image of the object A is 
formed at B, as before, but that the rays continuing 
onward are so refracted in passing through the lens C 
as to again cross, and thus, after further refraction by 
the lenses D and E, to form, at F, an erect image, 
which is magnified by the lens G. 

In both these figures, the limits of the page render 
it necessary to draw the angles of the rays very much 
out of proportion. 

288. Cross-Hairs. Since a considerable field of view 
is seen in looking through the telescope, it is necessary 
to provide means for directing the line of sight to the 
precise point which is to be observed. This could be 
effected by placing a very fine point, such as that of 
a needle, within the telescope, at some place where it 
could be distinctly seen. In practice, this fine point 
is obtained by the intersection of two very fine lines, 
placed in the common focus of the object-glass and 
of the eye-piece. These lines are called the cross- 
hairs, or cross-wires. Their intersection can be seen 
through the eye-piece, at the same time, and appar- 
ently at the same place, as the image of the distant 
object. The magnifying powers of the eye-piece will 
then detect the slightest deviation from perfect coin- 
cidence. " This application of the telescope may be 
considered as completely annihilating that part of the 
error of observation which might otherwise arise from 
an erroneous estimation of the direction in which an 
object lies from the observer's eve, or from the center of the in- 




THE INSTRUMENTS. 



191 



Fig. 203. 



Fig. 204. 



strument. It is, in fact, the grand source of all the precision of 
modern astronomy, without which all other refinements in in- 
strumental workmanship would be thrown away." What Sir 
John Herschei here says of its utility to astronomy is equally ap- 
plicable to surveying. 

The imaginary line which passes through the intersection of the 
cross-hairs and the optical center of the object-glass is called the 
line of collimation of the telescope.* 

The cross-hairs are attached to a ring, or short, thick tube of 
brass, placed within the 
telescope - tube, through 
holes in which pass loose- 
ly four screws, whose 
threads enter and take 
hold of the ring, behind 
or in front of the cross- 
hairs, as shown (in front 
view and in section) in the 
two figures in the margin. 
Their movements will be 
explained in " Adjust- 
ments." 

Usually, one cross-hair is horizontal, and the other vertical, as 
in Fig. 203,. but sometimes they are arranged as in 
Fig. 205, which is thought to enable the object to 
be bisected with more precision. A horizontal 
hair is sometimes added. 

The cross-hairs are best made of platinum wire, 
drawn out very fine by being previously inclosed 
in a larger wire of silver, and the silver then re- 
moved by nitric acid. Silk threads from a cocoon are sometimes 
used. Spiders' threads are, however, the most usual. If a cross- 
hair is broken, the ring must be taken out by removing two op- 
posite screws, and inserting a wire with a screw cut on its end, 
or a stick of suitable size, into one of the holes thus left open 

* From the Latin word collimo, or coUineo, meaning to direct one thing toward 
another in a straight line, or to aim at. The line of aim would express the meaning. 




Fig. 205. 




192 LAND-SURVEYim. 

in the ring, it being turned sidewise for that purpose, and then 
removing the other screws. The spiders' threads are then stretched 
across the notches seen in the end of the ring, and are fastened by 
gum, or varnish, or beeswax. The operation is a very delicate one. 
The following plan has been employed : A 
piece of wire is bent, as in the figure, so 



*G5^I 



J\j\/\l\l as to leave an opening a little wider than 
the ring of the cross-hairs. A cobweb is 
-^ chosen, at the end of which a spider is 

hanging, and it is wound around the bent 
wire, as in the figure, the weight of the insect keeping it tight 
and stretching it ready for use, each part being made fast by 
gum, etc. When a cross-hair is wanted, one of these is laid 
across the ring and there attached. One method is to draw the 
thread out of the spider, persuading him to spin, if he sulks, by 
tossing him from hand to hand. Another method is to unwind 
the spider-web from the cocoons, frequently to be found in spider- 
webs. A stock of such threads must be obtained in warm 
weather for the winter's wants. A piece of thin glass, with a 
horizontal and a vertical line etched on it, may be made a sub- 
stitute. 

289. Instrumental Parallax. This is an apparent movement 
of the cross-hairs about the object to which the line of sight is 
directed, taking place on any slight movement of the eye of the 
observer. It is caused by the image and the cross-hairs not being 
precisely in the common focus, or point of distinct vision of the 
eye-piece and the object-glass. To correct it, move the eye-piece 
out or in till the cross-hairs are seen clearly and sharply defined 
against any white object. Then move the object-glass in or out 
till the object is also distinctly seen. The cross-hairs will then 
seem to be fixed to the object, and no movement of the eye will 
cause them to appear to change their place. 

290. A milled-headed screw (on the farther side of the tele- 
scope, and not shown in the figure) passes into the telescope. 
and has a pinion at its other end entering a toothed rack (Fig. 



TEE INSTRUMENTS. 



193 



Fig. 207. 



oan/ 



207), and is used to move the object-glass, 0, out 
and in, according as the object looked at is 
nearer or farther than the one last observed. 
Short distances require a long tube ; long dis- 
tances a short tube. 

The eye-piece is moved in and out by a similar arrangement 
to the preceding. This movement is necessary in order to obtain 
a distinct view of the cross-hairs. Short-sighted persons require 
the eye-piece to be pushed farther in than persons of ordinary 
sight, and old or long-sighted persons to have it drawn far- 
ther out. 



291. Supports. The telescope of the transit is supported by a 
hollow axis at right angles to it, which itself rests, at each end, on 
two upright pieces, or standards, spreading at their bases so as to 
increase their stability. 

One end of the axis rests upon a movable block, which can be 
raised or lowered by a capstan-screw. The use of this will be 
shown in "Adjustments." 



Fig. 208. 



292. The Indexes. The supports, or standards, of the telescope 
just described are attached to the upper or index-carrying circle. 
This, as has been stated, can turn freely on the lower or graduated 
circle, by means of its conical axis moving in the hollow conical 
axis of the latter circle. This upper circle carries the index, V, 

which is an arrow-head or other mark 
on its edge, or the zero-point of a ver- 
nier scale. There are usually two of 
these, situated exactly opposite to each 
other, or at the extremities of a di- 
ameter of the upper circle, so that 
the readings on the graduated circle 
pointed out by them differ, if both 
are correct, exactly 180°. The object 
of this arrangement is to correct any 
error of eccentricity, arising from the center of the axis which 
carries the upper circle (and with which it and its index-pointers 




194 LAND-SURVEYING. 

turn), not being precisely in the center of the graduated circle. In 
the figure, let C be the true center of the graduated circle, but C 
the center on which the plate carrying the indexes turns. Let 
AC'B represent the direction of a sight taken to one object, 
and D' C E' the direction when turned to a second object. The 
angle subtended by the two objects at the center of the instru- 
ment is required. Let D E be a line passing through C, and 
parallel to D' E'. The angle A C D equals the required angle, 
which is therefore truly measured by the arc A D or B E. But 
if the arc shown by the index is read, it will be A D' on one 
side, and B E' on the other ; the first being too small by the arc 
D D', and the other too large by the equal arc E E'. If, how- 
ever, the half-sum of the two arcs AD' and BE' be taken, it 
will equal the true arc, and therefore correctly measure the an- 
gle. Thus, if AD' was 19°, and B E' 21°, their half-sum, 20°, 
would be the correct angle. , 

Three indexes, 120° apart, are sometimes used. They have the 
advantage of averaging the unavoidable inaccuracies and inequali- 
ties of graduation on diiferent parts of the limb, and thus dimin- 
ishing their effect on the resulting angle. 

293. The Graduated Circle. This is divided into three hun- 
dred and sixty equal parts, or degrees, and each of these is sub- 
divided into two or three parts or more, according to the size of the 
instrument. In the first case, the smallest division on the circle 
will of course be 30' ; in the second case 20'. More precise read- 
ing, to single minutes or even less, is effected by means of the ver- 
nier of the index, all the varieties of which will be fully explained 
under "Verniers." The numbers run from 0° around to 360°, 
which number is necessarily at the same point as the 0, or zero- 
point. In most instruments there is another concentric circle, on 
which the degrees are also numbered from 0° to 90°, as on the com- 
pass-circle. Each tenth degree is usually numbered, each fifth 
degree is distinguished by a longer line of division, and each de- 
gree-division line is longer than those of the subdivisions. A mag- 
nifying-glass is needed for reading the divisions with ease. In 
large instruments it is attached to each vernier. 



THE INSTRUMENTS. 195 

294. Movements. When the line of sight' of the telescope is 
directed to a distant, well-defined point, the nnaided hand of the 
observer can not move it with sufficient delicacy and precision to 
make the intersection of the cross-hairs exactly cover or " bisect " 
that point. To effect this, a clamp, and a tangent, or slow-motion, 
screw are required. This arrangement, as usually applied to the 
movement of the upper, or vernier plate, consists of a short post 
of brass, which is attached to the vernier-plate, and through 
which passes a long and fine-threaded " tangent-screw." The other 
end of this screw enters into and carries the clamp. This consists 
of two pieces of »brass, which, by turning the clamp-screw, which 
passes through them on the outside, can be made to take hold of and 
pinch tightly the edge of the lower circle, which lies between them 
on the inside. The upper circle is now prevented from moving on 
the lower one, for the tangent-screw keeps them at a fixed distance 
apart, so that they can not move to or from one another, nor con- 
sequently the two circles to which they are respectively made fast. 
But when this tangent-screw is turned by its milled head, it gives 
the clamp and with it the upper plate a smooth and slow motion, 
backward or forward, whence it is called the "slow-motion screw," 
as well as "tangent-screw," from the direction in which it acts. 
Another form of clamp is shown in Eig. 200. 

A little different arrangement is employed to give a similar 
motion to the lower circle on the body of the instrument. Its axis 
is embraced by a brass ring, into which enters a clamp-screw. 
The clamp-screw causes the ring to pinch and hold immovably 
the axis of the lower circle, while a turn of the tangent-screw 
will slowly move the clamp-ring itself, and therefore with it 
the lower circle. When the clamp is loosened, the lower circle, 
and with it everything above it, has a perfectly free motion. 

295. Levels. Since the object of the instrument is to measure 
horizontal angles, the circular plate on which they are measured 
must itself be made horizontal. , Whether it is so or not is known 
by means of two small levels placed on the plate at right angles to 
each other. Each consists of a glass tube, slightly curved upward 
in its middle, and so nearly filled with alcohol that only a small 



196 LAND-SURVEYIXG. 

bubble of air is left in the tube. This always rises to the highest 
part of the tubes. They are so " adjusted " that when this bubble 
of air is in the middle of the tubes, or its ends equidistant from the 
central mark, the plate on which they are fastened shall be level, 
which way soever it may be turned. One of the levels is some- 
times fixed between the standards above one of the verniers, and 
the other on the plate at the north end of the compass-box. 

296. Parallel Plates. To raise or lower either side of the circle, 
so as to bring the bubbles into the centers of the tubes, requires 
more gentle and steady movements than the unaided hands can 
give, and is attained by the parallel plates, and their four 
milled-headed screws, which hold the plates firmly apart, and, 
by being turned in or out, raise or lower one side or the other of 
the upper plate, and thereby of the graduated circle. The two 
plates are held together by a ball-and-socket joint. To level the 
instrument, loosen the lower clamp and turn the circle till each 
level is parallel to the vertical plane passing through a pair of 
opposite screws. Then take hold of two opposite screws and turn 

Fig. 209. 




them simultaneously and equally, but in contrary directions, screw- 
ing one in and the other out, as shown by the arrows in the figures. 
A rule easily remembered is that both thumbs must turn in, or 
both out. The movements represented in the first of these figures 
would raise the left-hand side of the circle and lower the right- 
hand side. The movements of the second figure would produce 
the reverse effect. Care is needed to turn the opposite screws 
equally, so that they shall not become so loose that the instrument 
will rock, or so tight as to be cramped. When this last occurs, 
one of the other pair should be loosened. 



THE INSTRUMENTS. 197 

Sometimes one of each pair of the screws is replaced by a strong 
spring, against which the remaining screws act. 

The French and German instruments, and most large instru- 
ments, are usually supported by only three screws. In such cases, 
one level is brought parallel to one pair of screws and leveled by 
them, and the other level has its bubble brought to its center by 
the third screw. If there is only one level on the instrument, it is 
first brought parallel to one pair of screws and leveled, and is then 
turned one quarter around so as to be perpendicular to them and 
ovpr the third screw, and the operation is repeated. 

297. Watch-Telescope. A second telescope is sometimes at- 
tached to the lower part of the instrument. When a number of 
angles are to be observed from any one station, direct the upper 
and principal telescope to the first object, and then direct the 
lower one to any other well-defined point. Then make all the 
desired observations with the upper telescope, and, when they are 
finished, look again through the lower one, to see that it and there- 
fore the divided circle have not been moved by the movements of 
the vernier-plate. The French call this the Witness- Telescope 
{Lunette temoiri). 

298. The Compass. Upon the upper plate is fixed a compass. 
It has been fully explained in Chapter III. It is little used in 
connection with the transit, which is so incomparably more accu- 
rate, except as a " check," or rough test of the accuracy of the 
angles taken, which should about equal the difference of the mag- 
netic bearings. 

299. The Reflector. In making observations on Polaris at night, 
or in surveying mines, a reflector (Fig. 210) is 

used. This is a silvered plate with a hole in it ^J 1G ' 210# 
for observing through with the telescope, while 
a light, held near the silvered surface, illuminates 
the cross-hairs. The reflector is attached to a 
ring, fitted to the object-glass slide, and is in- 
clined at an angle of 45° to the ring. 




198 



LAND-SUR VEYING. 



Fig. 211. 




300. The Diagonal Prism (Fig. 211). This is a prism attached 
to the eye-piece of the telescope, so that the 
rays of light, coming from the object sighted 
to, and passing through the telescope, are re- 
flected to the eye at an angle of 90° to the 
line of sight of the telescope. The prism is 
attached to a movable plate so that it can be 
turned to suit the position of the observer. 

This prism enables larger vertical angles to be 
measured than would be possible without it. 

The Transit. 

301. The Engineer's Transit (Fig. 213). This instrument is 
similar in general construction to that shown in Fig. 199, but 
differs from it in several important particulars. The sockets for 
the axes of the plates are longer and differently arranged. These 
are shown in Fig. 212. 

Both levels are attached to the upper plate. The verniers, in- 
stead of being placed at the sides between the legs of the standards. 

Fig. 212. 




as is usual, are placed near the north and south points of the 
compass-circle, so that the observer can read the vernier without 
stepping to the side of the instrument. The slow motion, both of 
the upper and lower plate, is given by one tangent-screw. In each 



THE INSTRUMENTS. 



199 



case an opposing spiral spring prevents any shake in the tangent- 
screw. 

The vertical arc is attached to the axis of the telescope by a 
clamp-screw, shown in the figure. The vernier and the slow- 
motion screw of the vertical arc are shown below the arc, and are 
attached to the left-hand standard. 



Fig. 21 




200 



LAND-SUB VEYING. 



Attached to the right-hand standard is the " Gradienter " 
(shown in detail in Fig. 245). 

302. A vertical section through the body of the engineer's 
transit is" given in Fig. 212. The lower plate, or " divided limb/' 
B, is supported by the hollow socket C. Through this hollow 
socket passes the conical spindle which supports the upper plate A. 

The upper plate carries the telescope, compass-box, and the 
verniers. The vernier-scales, V, V, are attached to the upper 
plate, but lie in the same plane as the divisions of the lower plate 
(so that the two can be viewed together without parallax), and are 
covered with glass to exclude dust. E is the clamp-screw. 



Fig. 214. 



303. The Theodolite. The transit, when furnished with a ver- 
tical circle and telescope level, is sometimes called a Theodolite. 
This name is used almost exclusively in England and on the Con- 
tinent of Europe. In one form of the theodolite the telescope can 
not be revolved on its horizontal axis. This 
form has been almost entirely superseded in this 
country by that having a reversible telescope. 
It is then called a Transit Theodolite, or simply 
a Transit. 

304. Goniasmometre. A very compact in- 
strument, to which this name has been given 
in France, where it is much used, is shown 
in the figure. The upper half of the cylinder 
is movable on its lower half. The observations 
may be taken through the slits, as in the sur- 
veyor's cross, or a telescope may be added to it. 
Readings may be taken both from the compass 
and from the divided edge of the lower half 
of the cylinder, by means of a vernier on the upper half.* 

* The proper care of instruments must not be overlooked. If varnished, they 
should be wiped gently with fine and clean linen. If polished with oil, they should 
be rubbed more strongly. The parts neither varnished nor oiled should be cleaned 
with Spanish-white and alcohol. Varnished wood, when spotted, should be wiped 
with very soft linen, moistened with a little olive-oil or alcohol. Unpainted wood is 




VERNIERS. 



201 



VERNIERS. 

S05. Definition. A vernier is a contrivance for measuring 
smaller portions of space than those into which a line is actually 
divided. It consists of a second line or scale, movable by the side 
of the first, and divided into equal parts, which are a very little 
shorter or longer than the parts into which the first line is divided. 
This small difference is the space which we are thus enabled to 
measure. * 

The vernier scale is usually constructed by taking a length 
equal to any number of parts on the divided line, and then divid- 
ing this length into a number of equal parts, one more or one less 
than the number into which the same length on the original line 
is divided. 

306. Illustration. The figure represents (to twice the real size) 
a scale of inches divided into tenths, with a vernier scale beside it, 
by which hundredths of an inch can be measured. The vernier is 

Fig. 215. 



Ost>©3^(^LO^^<3\?rH 



1 i 1 



V 



CS 



"t 



CO 



made by setting off on it nine tenths of an inch, and dividing that 
length into ten equal parts. Each space on the vernier is therefore 
equal to a tenth of nine tenths of an inch, or to nine hundredths 
of an inch, and is consequently one hundredth of an inch shorter 
than one of the divisions of the original scale. The first space of 
the vernier will therefore fall short of, or be overlapped by, the first 

cleaned with sand-paper. Apply olive-oil where steel rubs against brass ; and wax 
softened by tallow where brass rubs against brass. Clean the glasses with kid or 
buck skin. Wash them, if dirtied, with alcohol. 

* The vernier is so named from its inventor, in 1631. The name "Nonius," 
often improperly given to it, belongs to an entirely diiferent contrivance for a similar 
object. 



202 



LAND-SUR YEYING. 



space on the scale by this one hundredth of an inch ; the second 
space of the vernier will fall short by two hundredths of an inch ; 
and so on. If, then, the vernier be moved up by the side of the 
original scale, so that the line marked 1 coincides, or forms one 
straight line, with the line of the scale which was just above it, we 
know that the vernier has been moved one hundredth of an inch. 
If the line marked 2 comes to coincide with a line of the scale, the 



Fig. 216. 



7Y 



MINI 



t 



C^ 



t 



QQ 



vernier has moved up two hundredths of an inch ; and so for other 
numbers. If the position of the vernier be as in this figure, the 
line marked 7 on the vernier corresponding with some line on the 
scale, the zero-line of the vernier is seven hundredths of an inch 
above the division of the scale next below this zero-line. If this 
division be, as in the figure, 8 inches and 6 tenths, the reading will 
be 8-67 inches.* 

A vernier like this is used on some leveling-rods, being en- 
graved on the sides of the opening in the part of the target above 
its middle line. The rod being divided into hundredths of a foot, 
this vernier reads to thousandths of a foot. It is also used on some 
French mountain barometers, which are divided to hundredths of 
a metre, and thus read to thousandths of that unit. 

307. General Rules. To find what any vernier reads to — i. e. , 
to determine how small a distance it can measure — observe how 
many parts on the original line are equal to the same number in- 
creased or diminished by one on the vernier, and divide the length 



* The student will do well to draw such a scale aud vernier on two slips of thick 
paper, and move one beside the other till he can read them in any possible position ; 



and so with the following verniers. 



VERNIERS. 203 

of a part on the original line by this last number. It will give the 
required distance.* 

For verniers as usually constructed, the following rule will ap- 
ply : Divide the value of the smallest division on the original scale 
by the number of parts on the vernier. 

For example, if the limb of a transit be divided into half de- 
grees, and thirty parts on the vernier are equal to twenty-nine on 
the limb, then the value of the smallest division on the limb (30 
minutes), divided by the number of parts on the vernier (30) 
equals one minute. This is what the vernier reads to. 

To read any vernier, first, look at the zero-line of the vernier 
(which is sometimes marked by an arrow-head), and if it coincides 
with any division of the scale, that will be the correct reading, and 
the vernier divisions are not needed. But if, as usually happens, 
the zero-line of the vernier comes between any two divisions of the 
scale, note the nearest next less division on the scale, and then 
look along the vernier till you come to some line on it which ex- 
actly coincides, or forms a straight line, with some line (no matter 
which) on the fixed scale. The number of this line on the vernier 
(the 7th, in the last figure) tells that so many of the subdivisions 
which the vernier indicates are to be added to the reading of the 
entire divisions on the scale. 

When several lines on the vernier appear to coincide equally 
with lines of the scale, take the middle line. 

When no line coincides, but one line on the vernier is on one 
side of a line on the scale, and the next line on the vernier is as far 
on the other side of it, the true reading is midway between those 
indicated by these two lines. 

308. Retrograde Verniers. The spaces of the vernier in modern 
instruments are usually each shorter than those on the scale, a cer- 
tain number of parts on the scale being divided into a larger number 

* In algebraic language, let s equal the length of one part on the original line, 
and v the unknown length of one part on the vernier. Let m of the former = 

m 



m + 1 of the latter. Then ms = (m + 1 ) v. 



m + 1 



s — . If ms = ( m — 1) v, then v — s = — - — - 

m+l m + 1 v ; ' m— 1 

14 



204 



LAND-SUR VEYING. 



of parts on the vernier. * In the contrary case, f there is the incon- 
venience of being obliged to number the lines of the vernier and to 
count their coincidences with the lines of the scale, in a retrograde 
or contrary direction to that in which the numbers on the scale 
run. We will call such arrangements retrograde verniers. 

309. Illustration. In this figure, the scale, as before, repre- 
sents (to twice the real size) inches divided into tenths, but the 
vernier is made by dividing eleven parts of the scale into ten equal 

Fig. 217. 



\ 



O r* <Ti ffO ,.^ *° $3 {> C3 Cl> 2 



Y. 



i i i 



C^ 






parts, each of which is therefore one tenth of eleven tenths of an 
inch — i. e., eleven hundredths of an inch, or a tenth and a hun- 
dredth. Each space of the vernier therefore overlaps a space on 
the scale by one hundredth of an inch. The manner of reading 
this vernier is the same as in the last one, except that the num- 
bers run in a reverse direction. The reading of the figure is 
30-16. 

This vernier is the one generally applied to the common barom- 
eter, the zero-point of the vernier being brought to the level of 
the top of the mercury, whose height it then measures. It is also 
employed for leveling-rods which read downward from the middle 
of the target. 

310. Tig. 218 represents (to double size) the usual scale of 
the English mountain barometer. The scale is first divided 
into inches. These are subdivided into tenths by the longer 



* i. e., algebraically, v = 



+ 1 



f i. e., when v 



m- 1 



VERNIERS. 



205 



lines, and the shorter lines again divide these into half tenths, or 
to 5 hundredths ; 24 of these smaller parts are set off on the ver- 



Fig. 218. 



Up 



CO oi 



ITT LI I I 



1JLU I i i 1 I I l I i I I 



J L 



I I 



I L 



Mil illl 



V 



Mill 



I I 



C\2 



<m 



nier, and divided into 25 equal parts, each of which is therefore 

24 X '05 
= — grz — = '048 inch, and is shorter than a division of the scale 

by '050 — '048 = *002, or two thousandths of an inch, a twenty- 
fifth part of a division on the scale, to which minuteness the ver- 
nier can therefore read. The reading in the figure is 30*686 
(30*65 by the scale and *036 by the vernier), the dotted line marked 
D showing where the coincidence takes place. 

311. Circle divided into Degrees. The following illustrations 
apply to the measurements of angles, the circle being variously 
divided. In this article, the circle is supposed to be divided into 
degrees. 

If 6 spaces on the vernier are found to be equal to 5 on the 
circle, the vernier can read to one sixth of a space on the circle — 
i. e., to 10'. 

If 10 spaces on the vernier are equal to 9 on the circle, the 
vernier can read to one tenth of a space on the circle — i. e., to 6'. 

If 12 spaces on the vernier are equal to 11 on the circle, the 
vernier can read to one twelfth of a space on the circle — i. e., to 5'. 

Fig. 219 shows such an arrangement. The index, or zero, 
of the vernier is at a point beyond 358°, a certain distance, 
which the coincidence of the third line of the vernier (as indicated 
by the dotted and crossed line) shows to be 15'. The whole reading 
is therefore 358° 15'. 



206 



LAND-SUE VEYING. 



If 20 spaces on the vernier are equal to 19 on the circle, the 
vernier can read to one twentieth of a division on the circle — i. e. , 



Fig. 219. 



4 



10 



* 



60 



I 



60 45 30 15 




to 3'. English compasses, or " circumferentors," are sometimes 
thus arranged. 

If 60 spaces on the vernier are equal to 59 on the circle, the 
vernier can read to one sixtieth of a division on the circle — i. e., 
to 1'. 

312. Circle divided to 30'. Such a graduation is a very com- 
mon one. The vernier may be variously constructed. 



Fig. 220. 




VERMEILS. 



207 



Suppose 30 spaces on the vernier to be equal to 29 on the cir- 

29 X 30' 
cle. Each space on the vernier will be = — ^ — = 29', and will 

therefore 'be less than a space of the circle by 1', to which the ver- 
nier will then read. 

Fig. 220 shows this arrangement. The reading is 0°, or 360°. 

In Fig. 221 the dotted and crossed line shows what divisions 
coincide, and the reading is 20° 10' ; the vernier being the same 
as in the preceding figure, and its zero being at a point of the 
circle 10' beyond 20°. 

Fig. 221. 




In Fig. 222, the reading is 20° 40', the index being at a 
point beyond 20° 3', and the additional space being shown by 
the vernier to be 10'. 

Sometimes 30 spaces on the vernier are equal to 31 on the circle. 

31 X 30' 



Each space on the vernier will therefore be 



30 



= 31', and 



will be longer than a space on the circle by 1', to which it will 
therefore read, as in the last case, but the vernier will be " retro- 
grade." This is the vernier of the compass. The peculiar manner 
in which it is there applied is shown in Fig. 229. 



208 



LAND-SUE Y EYING. 
Fig. 222. 




If 15 spaces on the vernier are equal to 16 on the circle, each 

15 



1 fi v 3(Y 
space on the vernier will be = — ^ — = 32', and the vernier will 



therefore read to 2'. 

313. Circle divided to 20'. If 20 spaces of the vernier are 
equal to 19 on the circle, each space of the latter will be = 
19 X 20' 



20 



19', and the vernier will read to 20' — 19' = 1'. 



If 40 spaces on the vernier are equal to 41 on the circle, each 



Fig. 223. 




VERNIERS. 



209 



41 X 20' 
space on the vernier will be = — ^ — = 20£', and the vernier 

will therefore read to 20 J-' — 20' = 30". It will be retrograde. In 
Fig. 223 the reading is 360°, or 0° ; and it will be seen that the 



Fig. 224. 



50 



€ 



40 



10 9 8 SM6 5 4 3 2 



'* 



<f 



40 spaces on the vernier (numbered to whole minutes) are equal 

to 13° 40' on the limb — i. e., to 41 spaces, each of 20'. 

If 60 spaces on the vernier are equal to 59 on the circle, each 

59 X 20 
of the former will be = — ^- — = 19' 40", and the vernier will 

Fig. 225. 



20 



15 



10 



i-Xi I 



I II 



10 9 8 7 6 



I i i '__ 



5 4 3 2 1 1 



210 



LANB-SUR VEYING. 



therefore read to 20' — 19' 40" = 20". Fig. 224 shows such an 
arrangement. The reading in that position would be 40° 46' 20*. 

314. Circle divided to 15'. If 60 spaces on the vernier are 
equal to 59 on the circle, each space on the vernier will be = 
59 X 15' 



60 



14' 45", and the vernier will read to 15". In Fig. 225 



the reading is 10° 20' 45", the index pointing to 10° 15', and 
something more, which the vernier shows to be 5' 45". 



315. Circle divided to 10'. If 60 spaces on the vernier be 
equal to 59 on the limb,' the vernier will read to 10". In Fig. 
226 the reading is 7° 25' 40", the reading on the circle being 
7° 20', and the vernier showing the remaining space to be 5' 40". 



Fig. 226. 




10 987 6| 54321 



316. Reading backward. When an index carrying a vernier is 
moved backward, or in a contrary direction to that in which the 
numbers on the circle run, if we wish to read the space which it 
has passed over in this direction from the zero-point, the vernier 
must be read backward (i. e., the highest number be called 0), or 
its actual reading must be subtracted from the value of the small- 
est space on the circle. The reason is plain ; for, since the vernier 



VERNIERS. 



211 



shows how far the index, moving in one direction, has gone past 
one division-line, the distance which it is from the next division- 
line (which it may be supposed to have passed, moving in a con- 
trary direction) will be the difference between the reading and the 
value of one space. 

Thus, in Fig. 219, the reading is 358° 15'. But, counting back- 
ward from the 360°, or zero-point, it is 1° 45'. 

Caution on this point is particularly necessary in using small 
angles of deflection for railroad-curves. 

317. Arc of Excess. On the sextant and similar instruments, 
the divisions of the limb are carried onward a short distance be- 
yond the zero-point. This portion of the limb is called the " Arc 
of Excess." When the index of the vernier points to this arc, the 



f 



Fio. 227. 



5 







5 



i i i 



I ■• I 



ff 



I 



10 9 8 71 6 5 4 3 2 1 



-'I 



i 



reading must be made as explained in the last article. Thus, in 
the figure, the reading on the arc from the zero of the limb to the 
zero of the vernier is 4° 20', and something more, and the reading 
of the vernier from 10 toward the right, where the lines coin- 
cide, is 3' 20" (or it is 10' — 6' 40" = 3' 20'), and the entire reading 
is therefore 4° 23' 20". 

318. Double Verniers. To avoid the inconveniences of read- 
ing backward, double verniers are sometimes used. Fig. 228 



212 



LANDS UR VETING. 



shows one applied to a transit. Each of the verniers is like the 
one described in Art. 312, Figs. 220, 221, and 222. When the 
degrees are counted to the left, or as the numbers run, as is usual, 



Fig. 228. 



30j 20 | 10 ity i 10 




the left-hand vernier is to be read, as in Art. 312 ; but when the 
degrees are counted to the right, from the 360° line, the right- 
hand vernier is to be used. 



319. Oompass- Vernier. Another form of double vernier, often 
applied to the compass, is shown in Fig. 229. The limb is 



Fig. 229. 



— ^ 2|0 25 30 25 2,0 ^ 

3 |~To 5 4~T~j~J b 




divided to half-degrees, and the vernier reads to minutes, 30 
parts on it being equal to 31 on the limb. But the vernier is 
only half as long as in the previous case, going only to 15', the 
upper figures on one half being a sort of continuation of the lower 
figures on the other half. Thus, in moving the index to the right. 
read the loiver figures on the left-hand vernier (it being retrograde) 



ADJUSTMENTS. 213 

at any coincidence, when the space passed over is less than 15' ; 
but if it be more, read the upper figures on the right-hand vernier, 
and vice versa. 

ADJUSTMENTS. 

320. The purposes for which the transit (as well as most sur- 
veying and astronomical instruments) is to be used, require and 
presuppose certain parts and lines of the instrument to be placed 
in certain directions with respect to others ; these respective direc- 
tions being usually parallel or perpendicular. Such arrangements 
of their parts are called their Adjustments. The same word is also 
applied to placing these lines in these directions. In the following 
explanations the operations which determine whether these adjust- 
ments are correct, will be called their Verifications ; and the mak- 
ing them right, if they are not so, their Rectifications.* 

321. In observations of horizontal angles with the transit it is 
required — 

1. That the circular plates shall be horizontal in whatever way 
they may be turned around. 

2. That the telescope, when pointed forward, shall look in pre- 
cisely the reverse of its direction when pointed backward — i. e., 
that its two lines of sight (or lines of collimation) forward and 
backward shall lie in the same plane. 

3. That the telescope, in turning upward or downward, shall 
move in a truly vertical plane, in order that the angle measured 
between a low object and a high one may be precisely the same 
as would be the angle measured between the low object and a point 
exactly under the high object, and in the same horizontal plane as 
the low one. 

We shall see that all these adjustments are finally resolvable 
into these : 1. Making the vertical axis of the instrument perpen- 
dicular to the plane of the levels ; 2. Making the line of collima- 

* It has been well said that, " in the present state of science, it may be laid down 
as a maxim that every instrument should be so contrived that the observer may 
easily examine and rectify the principal parts ; for, however careful the instrument- 
maker may be, however perfect the execution thereof, it is not possible that any 
instrument should long remain accurately fixed in the position in which it came out 
of the maker's hands." (Adams's " Geometrical and Graphical Essays," 1791.) 



2U LAND-SURVEYING. 

tion perpendicular to its axis ; and, 3. Making this axis parallel to 
the plane of the levels. They are all best tested by the invaluable 
principle of "reversion." 

We have now, first, to examine whether these things are so — 
that is, to "verify" the adjustments; and, second, if we find 
that they are not so, to make them so — i. e., to "rectify" or "ad- 
just " them correctly. The above three requirements produce as 
many corresponding adjustments. 

322. First Adjustment. To cause the circle to le horizontal in 
every position. 

Verification. Turn the vernier-plate, which carries the levels, 
till one of them is parallel to one pair of the parallel plate-screws. 
The other will then be parallel to the other pair. Bring each bub- 
ble to the middle of its tube, by that pair of screws to which it is 
parallel. Then turn the vernier-plate half-way around — i. e., till 
the index has passed over 180°. If the bubbles remain in the cen- 
ters of the tubes, they are in adjustment. If either of them runs 
to one end of the tube, it requires rectification. 

Rectification. The fault which is to be rectified is that the 
plane of the level (i. e., the plane tangent to the highest point of 
the level tube) is not perpendicular to the vertical axis on which 

Fig. 230. Fig. 231. 

C 



the plate turns. For, let A B represent this plane, seen edgewise, 
and C D the center line of the vertical axis, which is here drawn as 
making an acute angle with this plane on the right-hand side. 
The first figure represents the bubble brought to the center of the 
tube. The second figure represents the plate turned half around. 
The center line of the axis is supposed to remain unmoved. The 



ADJUSTMENTS. 215 

acute angle will now be on the left-hand side, and the plate will no 
longer be horizontal ; consequently, the bubble will run to the 
higher end of the tube. The rectification necessary is evidently to 
raise one end of the tube and lower the other. The real error has 
been doubled to the eye by the reversion. Half of the motion of 
the bubble was caused by the tangent plane not being perpendicu- 
lar to the axis, and half by this axis not being vertical. Therefore, 
raise or lower one end of the level by the screws which fasten it to 
the plate, till the bubble comes about half-way back to the center, 
and then bring it quite back by turning its pair of parallel plate- 
screws. Then again reverse the vernier-plate 180°. The bubble 
should now remain in the center. If not, the operation should be 
repeated. The same must be done with the other level, if required. 
Then the bubbles will remain in the center during a complete revo- 
lution. This proves that the axis of the vernier-plate is then ver- 
tical ; and, as it has been fixed by the maker perpendicular to the 
plate, the latter must then be horizontal. 

It is also necessary to examine whether the bubbles remain in 
the center, when the divided circle is turned round on its axis. If 
not, the axes of the two plates are not parallel to each other. The 
defect can be remedied only by the maker ; for, if the bubbles be 
altered so as to be right for this reversal, they will be wrong for 
the vernier-plate reversal. 

323. Second Adjustment. To cause the line of collimation to 
revolve in a plane. 

Verification. Set up the transit in the middle of a level piece 
of ground, as at A in the figure. Level it carefully. Set a stake, 

Fig. 232. 




with a nail driven into its head, or a chain-pin, as far from the 
instrument as it is distinctly visible, as at B. Direct the telescope 



216 LAND-SURVEYING. 

to it, and fix the intersection of the cross-hairs very precisely upon 
it. Clamp the instrument. Measure from A to B. Then turn 
over the telescope, and set another stake at an equal distance from 
the transit, and also precisely in the line of sight. If the line of 
collimation has not continued in the same plane during its half- 
revolution, this stake will not be at E, but to one side, as at C. 
To discover the truth, loosen the clamp and turn the vernier-plate 
half around without touching the telescope. Sight to B, as at first, 
and again clamp it. Then turn over the telescope, and the line of 
sight will strike, as at D in the figure, as far to the right of the 
point as it did before to its left. 

Rectification. The fault which is to be rectified is that the 
line of collimation of the telescope is not perpendicular to the hori- 
zontal axis on which the telescope revolves. This will be seen by 

Fig. 233. 



B— - 



„,.. C 



the figures, which represent the position of the lines in each of the 
four observations which have been made. In each of the figures 
the long, thick line represents the telescope, and the short one the 
axis on which it turns. In Fig. 233 the line of sight is directed to 
B. In Fig. 234 the telescope has been turned over, and with it the 
axis, so that the obtuse angle marked in the first figure has 
taken the place, 0', of the acute angle, and the telescope points to 
C instead of to E. In Fig. 235 the vernier-plate Las been turned 



ADJUSTMENTS. 217 

half around so as to point to B again, and the same obtuse angle 
has got around to 0". In Fig. 236 the telescope has been turned 
over, the obtuse angle is at 0", and the telescope now points to D. 

To make the line of collimation perpendicular to the axis, the 
former must have its direction changed. This is effected by mov- 
ing the vertical hair the proper distance to one side. By loosen- 
ing the left-hand screw and tightening the right-hand one, the 
ring, and with it the cross-hairs, will be drawn to the right, and 
vice versa. Two holes at right angles to each other pass through 
the outer heads of the screws. Into these holes a stout steel wire 
is inserted, and the screws can thus be turned around. Screws so 
made are called "capstan-headed." One of the other pair of 
screws may need to be loosened to avoid straining the threads. In 
some French instruments, one of each pair of screws is replaced by 
a spring. 

To find how much to move this vertical hair, measure from C 
to D, Fig. 232 : Set a stake at the middle point E, and set another 
at the point F, midway between D and E. Move the vertical hair 
till the line of sight strikes F. Then the instrument is adjusted ; 
and, if the line of sight be now directed to E, it will strike B when 
the telescope is turned over, since the hair is moved half of the 
doubled error, D E. The operation will generally require to be 
repeated, not being quite perfected at first. 

It should be remembered that, if the telescope used does not 
invert objects, its eye-piece will do so. Consequently, with such a 
telescope, if it seems that the vertical hair should be moved to the 
left, it must be moved to the right, and vice versa. An inverting 
telescope does not invert the cross-hairs. 

If the young surveyor has any doubts as to the perfection of 
his rectification, he may set another stake exactly under the instru- 
ment by means of a plumb-line suspended from its center ; and 
then, in like manner, set his transit over B or E. He will find 
that the other two stakes, A and the extreme one, are in the same 
straight line with his instrument. 

In some instruments, the horizontal axis of the telescope can be 
taken out of its supports and turned over, end for end. In such a 
case, the line of sight may be directed to any well-defined point, 



218 LAND-SURVEYING. 

and the axis then taken out and turned over. If the line of sight 
again strikes the same point, this line is perpendicular to the axis. 



Fig. 237. 



^— % 



T 



If not, the apparent error is double the real error, as appears from 
the figures, the obtuse angle coming to 0', and the desired per- 
pendicular line falling at C midway between B and B'. The recti- 
fication may be made as before ; or, in some large instruments, in 
which the telescope is supported on Ys, by moving one of the Ys 
laterally. 

324. Third Adjustment. To cause the line of collimation to 
revolve in a vertical plane. 

Verification. Suspend a long plumb-line from some high point. 
Set the instrument near this line, and level it carefully. Direct 
the telescope to the plumb-line, and see if the intersection of the 
cross-hairs follows and remains upon this line when the telescope 
is turned up and down. If it does, it moves in a vertical plane. 

The angle of a new and well-built house will form an imperfect 
substitute for the plumb-line. 

Otherwise : The instrument being set up and leveled as above, 
place a basin of some reflecting liquid (quicksilver being the best, 
though molasses, or oil, or even water will answer, though less per- 
fectly) so that the top of a steeple, or other point of a high object, 
can be seen in it through the telescope by reflection. Make the 
intersection of the cross-hairs cover it. Then turn up the tele- 
scope, and, if the intersection of the cross-hairs bisects also the 
object seen directly, the line of sight has moved in a vertical plane. 
If a star be taken as the object, the star and its reflection will be 



ADJUSTMENTS. 219 

equivalent (if it be nearly overhead) to a plumb-line at least fifty 
million million miles long. 

Otherivise : Set the instrument as close as possible to the base 
of a steeple or other high object ; level it, and direct it to the top 
of the steeple, or to some other elevated and well- 
defined point. Clamp the plates. Turn down the FlQ - 239. 
telescope, and set up a pin in the ground precisely "in 
line." Then loosen the clamp, turn over the telescope, 
and turn it half-way around, or so far as to again 
sight to the high point. Clamp the plates, and again 
turn down the telescope. If the line of sight again 
strikes the pin, the telescope has moved in a vertical I 
plane. If not, the apparent error is double the real / 
error. For, let S be the top of the steeple (Fig. 239), y 
and P' the pin ; then the plane in which the telescope 
moves, seen edgewise, is S P' ; and, after being turned around, 
the line of sight moves in the plane S P", as far to one side of 
the vertical plane S P as S P' was on the other side of it. 

Rectification. Since the second adjustment causes the line of 
sight to move in a plane perpendicular to the axis on which it 
turns, it will move in a vertical plane if that axis be horizontal. 
It can be made so by raising or lowering one end of the axis by 
means of a screw placed in the standard for that purpose. 

325. Centering Eye-Piece. In some instruments, such as that 
of which a longitudinal section is shown in the margin, the inner 
end of the eye-piece may be moved so that the cross-hairs shall be 
seen precisely in the center of its field of view. This is done by 
means of four screws, arranged in pairs, like those of the cross-hair 
ring-screws, and capable of moving the eye-piece up and down, and 
to right or left, by loosening one and tightening the opposite one. 
Two of them are shown at A, A, in the figure, in which B, B, are 
two of the cross-hair screws. 

326. Centering Object-Glass. In some instruments four screws, 

similarly arranged, two of which are shown at C, C, can move, in 

any direction, the inner end of the slide which carries the object- 
15 



220 



LAND-SUE VEYINO. 



glass. 



Fig. 240 




The necessity for such an arrangement arises from the 
impossibility of drawing a tube perfectly 
straight. Consequently, the line of collima- 
tion, when the tube is drawn in, will not 
coincide with the same line when the tube 
is pushed out. If adjusted for one position, 
it will therefore be wrong for the other. 
These screws, however, can make it right 
in both positions. They are used as fol- 
lows : 

Sight to some well-defined point as far off 
as it can be distinctly seen. Then, having 
the plates firmly clamped, move out the ob- 
ject-glass slide, and fix a point in the line of 
sight as close to the instrument as can be 
distinctly seen. Then turn the limb half- 
way around horizontally, reverse the tele- 
scope, and again sight to the near point, by 
clamping the plates and bringing the verti- 
cal cross-hair on the point by means of the 
tangent-screw. Then draw in the object- 
glass slide until the distant object is distinct- 
ly seen. If the vertical cross-hair bisects it, 
no adjustment is necessary. If not, correct 
one half of the apparent error by means of 
the screws C C in Fig. 240. This may dis- 
turb the second adjustment. Try that over 
again, and again perform the operation of 
centering the object-glass. 

This adjustment is always performed by 
the maker, and its screws are covered by a 
short tube. 

All the adjustments should be meddled 
with as little as possible, lest the screws 
should get loose ; and, when once made 
right, they should be kept so by careful 
usage. 




ADJUSTMENTS. 221 

327. Fourth Adjustment. To cause the line of collimation of the 
telescope to be horizontal when the bubble of the level attached to it 
is in the center of its tube. 

Drive two pegs several hundred feet apart, and set the instru- 
ment midway between them. Level, and sight to the rod held on 
each peg. The difference of the readings will be the true differ- 
ence of the heights of the pegs, no matter how much the level 
may be out of adjustment. 

Then set the instrument over one peg, and sight to the rod held 
at the other. Measure the height of the cross-hairs above the first 
peg. The difference of this height and the reading on the rod 
should equal the difference of the heights of the two points, as 
previously determined. If it does not, set the target to the sum or 
difference of the height of, the cross-hairs above the first peg and 
the true difference of height of the points, according as the first 
point is higher or lower than the second, and hold the rod on the 
second point. Sight to it, and raise or lower one end of the bubble- 
tube until the horizontal cross-hair does bisect the target when the 
bubble is in the center. 

Instead of setting over one peg, it is generally more convenient 
to set near to it, and sight to a rod held on it, and use this reading 
instead of the measured height of the cross-hairs. 

328. Fifth Adjustment. To make the vernier of the vertical 
circle read zero when the bubble of the telescope - level is in the 
center. 

This is verified in various ways : 

1. By simple inspection. 

2. By reversion. Sight to some point. Note the reading on 
the vertical circle. Turn the telescope half-way around horizon- 
tally. Turn over the telescope and again observe the same point, 
and note the reading. Half the difference (if any) of the two read- 
ings is the error. 

3. By reciprocal observations. Observe successively from each 
of two points to the other. Half the difference of the readings 
equals the index-error. 

When the verification has been made, the error may be rectified 



Fig. 241. 



222 LAND-SURVEYING. 

on the instrument by moving the vernier-plate, or the circle, or 
noted as a correction to each observation when the instrument is 
large and delicate. 

THE FIELD-WORK. 

329. To measure a Horizontal Angle. Set up the instrument 
so that its center shall be exactly over the angular point, or in the 
intersection of the two lines whose difference of direction is to be 
measured, as at B in the figure. A plumb-line must be suspended 
from under the center. Dropping a stone is an imperfect substi- 
tute for this. Set the instru- 
ment so that its lower paral- 
lel plate may be as nearly hor- 
izontal as possible. The lev- 
AO — ^3 e ^ s w ^ serYe as guides if the 

four parallel-plate screws be 
first so screwed up or down that equal lengths of them shall be 
above the upper plate. Then level the instrument carefully. 
Direct the telescope to a rod, stake, or other object, A in the 
figure, on one of the lines' which form the angle. Tighten the 
clamps, and by the tangent-screw move the telescope so that the 
intersection of the cross-hairs shall very precisely bisect this ob- 
ject. Note the reading of the vernier. Then loosen the clamp 
of the vernier, and direct the telescope on the other line (as to C) 
precisely as before, and again read. The difference of the two 
readings will be the desired angle, ABC. Thus, if the first read- 
ing had been 40° and the last 190°, the angle would be 150°. If 
the vernier had passed 360° in turning to the second object, 360° 
should be added to the last reading before subtracting. Thus, if 
the first reading had been 300°, and the last reading 90°, the an- 
gle would be found by calling the last reading, as it really is, 
360° + 90° = 450°, and then subtracting 300°. 

It is best to sight first to the left-hand object and then to the 
right-hand one, turning "with the sun" or like the hands of a 
watch, since the numbering of the degrees usually runs in that 
direction. 

It is convenient, though not necessary, to begin by setting the 



TEE FIELD-WORK. 223 

yernier at zero by the upper movement (that of the vernier-plate 
on the circle), and then, by means of the lower motion (that of the 
whole instrument on its axis), to direct the telescope to the first 
object. Then fasten the lower clamp, and sight to the second 
object as before. The reading will then be the angle desired. An 
objection to this is that the two verniers seldom read alike.* 

After one or more angles have been observed from one point, 
the telescope must be directed back to the first object, and the 
reading to it noted, so as to make sure that it has not slipped. A 
watch-telescope renders this unnecessary. 

The error arising from the instrument not being set precisely 
over the center of the station will be greater the nearer the object 
sighted to. Thus, a difference of one inch would cause an error of 
only 3" in the apparent direction of an object a mile distant, but 
one of nearly 3' at a distance of a hundred feet. 

330. Reduction of High and Low Objects. When one of the 
objects sighted to is higher than the other, the "plunging tele- 
scope " of this instrument causes the angle measured to be the 
true horizontal angle desired — i. e., the same angle as if a point 
exactly under the high object and on a level with the low object 
(or vice versa) had been sighted to. For the telescope has been 
caused to move in a vertical plane by the third adjustment, and 
the angle measured is therefore the angle between the vertical 
planes which pass through the two objects, and which "project" 
the two lines of sight on the same horizontal plane. 

This constitutes the great practical advantage of these instru- 
ments over those which are held in the planes of the two objects 
observed, such as the sextant and the "circle," much used by the 
French. 

331. Notation of Angles. The angles observed may be noted 
in various ways. Thus, the observation of the angle ABC, in 

* The learner will do well to gauge his own precision and that of the instrument 
(and he may rest assured that his own will be the one chiefly in fault) by measuring, 
from any station, the angles between successive points all around him, till he gets 
back to the first point, beginning at different parts of the circle for each angle. The 
sum of all these angles should exactly equal 360°. He will probably find quite a 
difference from that. 



224. LAND-SURVEYIXQ. 

Fig. 241, may be noted "At B, from A to C, 150°," or, better, 
"At B, between A and C, 150°." In column form, this becomes 
Between A150 c jand C. 
At j B I 

When the vernier had been set at zero before sighting to the 
first object, and other objects were then sighted to, those objects, 
the readings to which were less than 180°, will be on the left of the 
first line, and those to which the readings were more than 180° will 
be on its right, looking in the direction in which the survey is pro- 
ceeding, from A to B, and so on. 

In surveying a farm, the angle of deflection at station, or the 
traverse angle, may be noted, together with the lengths of the 
courses. 

332. To repeat an Angle. Begin as in Art. 329, and measure 
the angle as there directed. Then unclamp below, and turn the 
circle around till the telescope is again directed to the first object, 
and made to bisect it precisely by the lower tangent-screw. Then 
unclamp above and turn the vernier-plate till the telescope again 
points to the second object, the first reading remaining unchanged. 
The angle will now have been measured a second time, but on a 
part of the circle adjoining that on which it was first measured, 
the second arc beginning where the first ended. The difference be- 
tween the first and last readings will therefore be twice the angle. 

This operation may be repeated a third, a fourth,, or any num- 
ber of times, always turning the telescope back to the first object 
by the lower movement (so as to start with the reading at which 
the preceding observation left off), and turning it to the second 
object by the upper movement. Take the difference of the first 
and last readings and divide by the number of observations. 

The advantage of this method is that the errors of observation 
(i. e., sighting sometimes to the right and sometimes to the left of 
the true point) balance each other in a number of repetitions, 
while the constant error of graduation is reduced in proportion to 
this number. This beautiful principle has some imperfections in 
practice, probably arising from the slipping and straining of the 
clamps. 



Fig. 242. 




THE FIELD-WORK. 225 

333. Angles of Deflection. The angle of deflection of one line 
from another is the angle which one line makes with the other line 
produced. Thus, in the 
figure, the angle of deflec- 
tion of B C from A B is 
B'BC. It is evidently the 
supplement of the angle 
ABO. 

To measure it with the Transit, set the instrument at B, direct 
the telescope to A, and then turn it over. It will now point in the 
direction of A B produced, or to B', if the second adjustment has 
been performed. Note the reading. Then direct the telescope to 
C. Note the new reading, and their difference will be the re- 
quired angle of deflection, B' B 0. 

If the vernier be set at zero before taking the first observation, 
the readings for objects on the right of the first line will be less 
than 180°, and more than 180° for objects on the left, conversely 
to Art, 331. 

334. Line - Surveying. The survey of a line, such as a road, 
etc., can be made by the transit with great precision, measuring 
the angle which each line makes with the preceding line, and 
noting their lengths, and the necessary offsets on each side. 

Short lines of sight should be avoided, since a slight inaccuracy 
in setting the center of the instrument exactly over or under the 
point previously sighted to would then much affect the angle. 
Very great accuracy can be obtained by using three tripods. One 
would be set at the first station and sighted back to from the 
instrument placed at the second station, and a forward sight be 
then taken to the third tripod placed at the third station. The 
instrument would then be set on this third tripod, a back-sight 
taken to the tripod remaining on the second station, and a fore- 
sight taken to the tripod brought from the first station to the 
fourth station, to which the instrument is next taken, and so on. 
This is especially valuable in surveys of mines. 

The field-notes may be taken as directed in compass-surveying, 
the angles taking the place of the bearings. The "checks by 



226 LAND-SURVEYING. 

intersecting bearings/' before explained, should also be employed. 
The angles made on each side of the stations may both be meas- 
ured, and the equality of their sum to 360° would at once prove 
the accuracy of the work. 

If the magnetic bearing of any one of the lines be given, and 
that of any of the other lines of the series be required, it can be 
deduced by constructing a diagram, or by modifications of the 
rules given for the reverse object. 

335. Traversing; or, surveying by the Back-Angle. This is a 
method of observing and recording the different directions of suc- 
cessive portions of a line 

"Ftp 24^ 

' ' , (such as a road, the bound- 

A B s\^ 

— .^- ar - es Q £ a f arm e tc.) so as 

c / \ in 

to read off on the instru- 




ment, at each station, the 
angle which each line 
makes — not with the pre- 
ceding line— but with the first line observed, or some other con- 
stant line. This line is, therefore, called the meridian of that sur- 
vey. 

The operation consists essentially in taking -each back-sight by 
the lower motion (which turns the circle without changing the 
reading), and taking each forward sight by the upper motion, 
which moves the vernier over the arc measuring the new angle ; 
and thus adds it to or subtracts it from the previous reading. 

Set up the instrument at some station, as B ; put the vernier at 
zero, and, by the lower motion, sight back to A. Tighten the 
lower clamp, reverse the telescope, loosen the upper clamp, sight 
to C by the upper motion, and clamp the vernier-plate again. Re- 
move the instrument to C, sight back to B by the lower motion. 
Then clamp below, reverse the telescope, loosen the upper clamp, 
and sight to D by the upper motion. Then go to D and pro- 
ceed as at ; and so on. The reading gives the angles measured 
to the right or "with the sun," as shown by the arcs in the 
figure. 

Care should be taken to keep the same side of the instrument 



THE FIELD-WORK. 227 

ahead, and, if only one vernier is read, to read from the same ver- 
nier. 

The chief advantage of this method is its greater rapidity in 
the field and in platting, the angles being all laid down from one 
meridian, as in compass-surveying. 

336. Use of the Compass. The chief use of the compass at- 
tached to a transit is as a check on the observations ; for the 
difference between the magnetic bearings of any two lines should 
be the same, approximately, as the angle between them, measured 
by the more accurate instruments. The bearing also prevents any 
ambiguity as to whether an angle was taken to the right or to 
the left. 

The instrument may also be used like a simple compass, the 
telescope taking the place of the sights, and requiring similar tests 
of accuracy. A more precise way of taking a bearing is to turn 
the plate to which the compass-box is attached, till the needle 
points to zero, and note the reading of the vernier ; then sight to 
the object, and again read the vernier. The bearing will thus be 
obtained more minutely than the divisions on the compass-box 
could give it. 

337. Ranging out Lines. This is the converse of - surveying- 
lines. The instrument is fixed over the first station with great 
precision, its telescope being very carefully adjusted to move in a 
vertical plane. A series of stakes, with nails driven in their tops, 
or otherwise well defined, are then set in the desired line as far as 
the power of the instrument extends. It is then taken forward to 
a stake three or four from the last one set, and is fixed over it, 
first by the plumb and then by sighting backward and forward to 
the first and last stake. The line is then continued as before. A 
good object for a long sight is a board painted like a target, with 
black and white concentric rings, and made to slide in grooves cut 
in the tops of two stakes set in the ground about in the line. It 
is moved till the vertical hair bisects the circles (which the eye 
can determine with great precision), and a plumb-line dropped 
from their center gives the place of the stake. " Mason and Dix- 
on's Line" was thus ranged. 



228 LAND-SUR VE YING. 

When the transit is used for ranging, its " Second Adjust- 
ment " is most important, to insure the accuracy of the reversal of 
its telescope. 

338. Farm-Surveying, etc. A farm can be much more accu- 
rately surveyed with the transit than with the compass. The farm 
should be kept on the right hand, and then the angles measured 
will be the supplements of the interior angles. If the angles to the 
right be called positive, and those to the left negative, their alge- 
braic sum should equal 360°. 

If the boundary-lines be surveyed by " Traversing," the read- 
ing, on getting back to the last station and looking back to the 
first line, should be 360°, or 0°. 

The content of any surface surveyed by ' ' Traversing " with the 
transit can be calculated by the traverse-table, by the following 
modification : When the angle of deflection of any side from the 
first side, or meridian, is less than 90°, call this angle the bearing, 
find its latitude and departure, and call them both plus. When 
the angle is between 90° and 180°, call the difference between the 

Fig. 244. 



89i° 7® 



THE FIELD-WORK. 



229 



angle and 180° the bearing, and call its latitude minus and its de- 
parture plus. When the angle is between 180° and 270°, call its 
difference from 180° the bearing, and call its latitude minus and 
its departure minus. When the angle is more than 270°, call its 
difference from 360° the bearing, and call its latitude plus and its 
departure minus. Then use these as in getting the content of a 
compass-survey. The signs of the latitudes and departures follow 
those of the cosines and sines in the successive quadrants. 

Fig. 244 is a plat of the survey worked out in Art. 255. 

The following table gives the deflection angle at each station, 
the traverse angle (i. e., the angle which each line makes with the 
first one), and the reduced bearing, calling the first line (1 to 2) 
the meridian : 



STATIONS. 


DEFLECTION ANGLES. 


TRAVERSE ANGLES. 


BEARINGS. 


1 

2- 
' 3 
4 
5 


9H° 
48|° 
39J° 
911° 
89i° 


0° or 360° 

48i° 

88° 
179i° 
268i° 


North. 
N. 48i° E. 
N. 88° E. 
S. |°E. 
S. 88i° W. 



If the deflection angle at station 1 (91J°) be added to the trav- 
erse angle at station 5, the sum will be 360°. 

Any side may be taken as the meridian of the survey. 

If the true bearing of one side be known, the true bearings of 
the other sides may be determined by Art. 189. 

The content is calculated by latitudes and departures, as in 
compass-surveying. 

The latitudes and departures may be taken from the tables, in- 
terpolating for minutes as in Art. 242, or they may be calculated 
with a table of natural sines and cosines, as in Art. 240. 



Example 



FIELD-BOOK. 



STATIONS. 


ANGLES OF DEFLECTION. 


DISTANCES IN CHAINS. 


1 

2 
3 
4 
5 


62° 15' 

86° 38' 
59° 20' 
80° 6' 
71° 41' 


4-64 
3-60 
4-15 
4-22 
3-25 



230 



LAND-SUR VEYING. 



CALCCXATION OF AEEAS, CALLING COURSE 1 TO 2 THE HEEIDIAN, AND USING 
SINES AND COSINES INSTEAD OF A TEAVEESE TABLE. 



03 

O 

< 
Eh 
W 


BEARINGS. 


H 
O 

5 

CO 

Q 


co 

g 

S3 


to 
a 
2; 

CO 

o 


LATI- 
TUDES. 


DEPART* 
URES. 


i 

DOUBLE 
LOXGI- 
TTJDBS. 


DOUBLE AREAS. 


+ | — + 


- 


+ 


- 


1 

2 
3 
4 
5 


+ 00°, 00' + 

+ 86°, 38' + 
-34°, 2' + 
-46°, 4'- 
+ 62°, 15'- 


4-64 
3-60 
4-15 
4-22 
325 


•ooooo 

•99827 
•55968 
•72015 
•88499 


1-00000 
•05873 
•82871 
•69382 
•46561 


4'64 
0-21 

1-52 

6-37 


o-oo 

I3-59 
3-44 2-32 
2-931 

6-375-91 


3-04 

2-87 

5-91 


o-oo 

+ 3-59 
+ 9-50 
+ 8-78 
+ 2-87 


o-oooo 

•7539 
4-3624 


32-6800 
25-7254 


5-1163 


58-4054 
5-1163 




2)53-2891 
















Square chains, 


26-6445 



339. When the lengths of the sides are measured with an engi- 
neer's chain, and the distances are determined in feet, the process 
of calculating the area is the same as for chains and decimals. The 
area is obtained in square feet instead of square chains, and to 
reduce it to acres it will be necessary to divide by 43560, the num- 
ber of square feet in an acre. 

340. Platting. Any of these surveys can be platted by any of 
the methods explained and characterized in Chapter III. A circu- 
lar protractor may be regarded as a theodolite placed on the paper. 
' ' Platting Bearings" can be employed when the survey has been 
made by "Traversing." But the method of "Latitudes and de- 
partures " is by far the most accurate. 



THE GRADIENTER. 

311. This is an attachment to the transit for determining grades 
and distances. It consists of an arm, attached to the axis of the 
telescope, and a micrometer-screw, by means of which the move- 
ment of the arm, and consequently of the telescope, can be accu- 
rately measured. 

The arm is placed on the inside of one of the standards, and is 
attac led to the telescope axis by means of a clamp-screw, so that 
it may be clamped or loosened at pleasure. 



THE GRADIENTER. 



231 



The method of measuring the movement of the arm is shown 
in Fig. 245. 

C is a section of the axis of the telescope. B is the arm, which 



Fig. 215. 




is clamped to the axis by the screw D. M is the micrometer-screw. 
A is a lip projecting from a plate fastened to the standards. 

The screw is accurately cut, so that one revolution of the screw 
will cause the horizontal cross-hair of the telescope to move over a 
given space (say one foot) on a rod held at a given distance, as 100 
feet. The head of the screw is graduated into equal parts, usually 
50 or 100. Above the graduated head is a scale so graduated that 
one revolution of the screw will move the head over one space on 
the scale. Thus the number of whole revolutions given to the 
screw may be read on the scale, and the parts of a revolution read 
on the graduated head. 

The point of the screw presses against the lip, A, and is held 
firmly against it by the opposing spiral spring, S. 

When the arm is made fast to the axis by the clamp-screw, D, 
and the gradienter-screw, M, is turned, it will turn the telescope 
vertically on its axis, and the distance which the horizontal cross- 



232 LAND-SURVEYING. 

hair will pass oyer on a rod, toward which the telescope is pointed, 
will vary directly with the distance from the transit to the rod. 

342. To establish Grades. Let us suppose that one revolution 
of the gradienter-screw will move the horizontal cross-hair over a 
space of one foot, on a rod held at a distance of 100 feet from the 
transit. Then, to set grades, we have only to level the telescope, 
clamp the gradienter-arm, and turn the micrometer-screw through 
as many divisions of the head (graduated into 100 parts) as there 
are hundredths of a foot rise or fall per hundred feet of horizontal 
distance ; raising the cross-hair for an up-grade, and lowering it 
for a down-grade. The line of sight will then be on the required 
grade. 

If the transit be set over a point of the required grade-line, set 
the target on the rod at the height of the center of the telescope- 
axis above the given point, and then the bottom of the rod, held 
at any point on the line, will be at a point in the desired grade- 
line when the horizontal cross-hair bisects the target. 

Thus, if the grade is to be 1-64 feet per hundred, turn the 
screw one entire revolution and 64 of the divisions on the gradu- 
ated head, and the line of sight will then be on the required grade. 

343. To measure Distances. When the ground is level or ap- 
proximately so, see what space on the rod the horizontal cross-hair 
moves over for one revolution of the gradienter-screw. Then the 
distance in feet will be equal to the space on the rod, expressed in 
feet and decimals, multiplied by 100. 

Thus, if the space on the rod, moved over by the cross-hair 
„ aMe for one revolution of 

Fig. 246. 

E the gradienter - screw, 
was 4-27 feet, the dis- 
tance at which the rod 
was held was 427 feet. 
For, in Fig. 246, let 
A be the position of the transit ; C B, the reading on the rod, held 
at a distance of 100 feet, for one revolution of the screw ; and D E 
the space passed over on the rod for one revolution of the screw 



c 



TEE QUADIENTEE. 



233 



when the rod is held at the unknown distance AD. It is evi- 
dent that the triangles ABO and ADE are similar, and that 
CB:AB::ED:AD, 
or, 1 : 100 :: 4*27 : 427. 
If the rod sighted to is only graduated to feet— as an ordinary 
transit-rod— find how many revolutions and parts of revolutions 
will move the horizontal cross-hair over a whole number of feet on 
the rod. Then, since one revolution of the screw will move the 
cross-hair over a space of one foot on the rod at a distance of 100 
feet, we have the proportion : As the number of revolutions of the 
screw (whole numbers and decimals) is to 100 feet, so is the num- 
ber of feet passed over on the rod by the cross-hair to the required 
distance. For, from Fig. 246 we have, as before : 
CB: AB :: DE: AD. 
O B now represents what the reading on the rod (in feet and 
decimals), held at a distance of 100 feet, would be for the given 
number of revolutions : A B is 100', D E is the reading on the rod 
in feet, and AD is the required distance. 

Suppose, for example, the gradienter-screw be turned 1*25 
time, and the space passed over on the rod by the cross-hair be 3 
feet. Then we have : 

1-25 : 100 :: 3 : 240. 
. \ The required distance is 240 feet. 

Problem. — When no graduated rod is available, to determine a distance 
by using, in place of a rod, a stick whose length can afterward be measured. 

On sloping ground, the methods given will apply, if the rod be 

held perpendicular to the line of sight. This, however, is not 

easily done. It will 

, , ,. , , Fig. 247. 

be better to apply q 

methods specially ^-"f 

adapted to sloping 

ground. 

344. On Sloping 
Ground. In Fig. 

247, let A be the 
position of the tran- 




234 



LAND-SUR VETING. 



sit ; G- the point over which it is set ; C where the rod is held ; 

A B a horizontal line through the axis of the telescope ; A C 

the distance from the horizontal axis of the telescope to the 

foot of the rod ; and C D the distance, on a vertical rod, 

passed over by the horizontal cross-hair for one revolution of 

the gradienter-screw. Let F be perpendicular to A 0, and 

D B to A B. 

Represent the angle of elevation, B A C, by e, the angle A D 

by s, and the distance D C by fa Then we have : 

DB = DC + CB. 

. \ A B tan. {s + e) = h + A B tan. e, 

h 

and A B = -, — : — r . 

tan. (s + e) — tan. e 

For convenience of computation, this may be put in another 

form. Add and subtract 100 k, and we have : 

AB=100&- 100 &+ r j—tA 1 • 

tan. (s -{- e) — tan. e 

And, since tan. s = jfa, 
AB = 100 h — Jc (100 sin. e -j- cos. e) sin. e. 

The quantity (100 sin. e + cos. e) 
sin. e, for angles from 1° to 20° will 
be found in the table for the gradi- 
enter. Hence the rule : 

Multiply the rod-reading by 100, 
and deduct the product of the rod- 
reading by the tabular number corre- 
sponding to the angle of elevation, e. 
The result will be the horizontal dis- 
tance A B. 

Example. Angle of elevation, 4° ; 
rod-reading, 2-63 feet. 

2-63 X 100 = 263 
2-63 X .5= 1-3 



TABLE FOE GEADIENTEE 



ANGLE OF 


(100 sin. e + cos.e) 


ELEVATION. 


x siN.e. 


0° 


•o 


1° 


•1 


2° 


•2 


3° 


•3 


4° 


•5 


5° 


•8 


6° 


1-2 


7° 


1-6 


8° 


2-1 


9° 


2'6 


10° 


3-2 


11° 


3-8 


12° 


4-5 


13° 


5-3 


14° 


6-1 


15° 


7-0 


16° 


7-9 


17° 


8-8 


18° 


9-8 


19° 


10-9 


20° 


12-0 



Horizontal distance, 261 *7 
The table for the correction is com- 
puted to tenths only, as the unavoid- 



THE STADIA OR TELEMETER. 235 

able errors in using the instrument would render any more exact 
computation useless. 

For ordinary cases, when the angle of elevation is small, the 
computation for the distance and correction can be made mentally. 

345. The horizontal distance, AB, is the one almost always 
required, as all measurements of distances in surveying and engi- 
neering should be made horizontally. 

The distance from the transit to the point at which the rod is 
held (i. e., AC) is equal to the horizontal distance, AB, divided 
by cos. e. 

The distance G may be found by solving the triangle C A G, 
of which the sides A G and A C, and the included angle GAG, are 
known. 

When the angle e is an angle of depression, the top of the rod 
is taken for the point c, and the distance C D is measured down- 
ward from the top of the rod. 

In using the micrometer-screw, care must be taken, when meas- 
uring, to always turn the screw in the same direction, in order 
to avoid any lost motion in the screw. In determining the space 
passed over by the cross-hair for one revolution of the screw, set 
the screw back of the first reading, and bring it up by turning the 
screw in the same direction in which it is to be turned for making 
the measurement. 

THE STADIA OB TELEMETER. 

346. On the cross-hair ring of the telescope stretch two more 
horizontal cross-hairs of spider-web or platinum wire, at equal 
distances above and below the original one. The two additional 
wires are called Stadia Wires. The stadia wires may be either 
fixed or adjustable. In the former case they may be attached 
directly to the cross-hair ring. When they are adjustable, each 
may be fastened to a separate slide, actuated by a capstan-screw on 
the outside of the telescope-tube, as shown in Figs. 248 and 249. 

The slides to which the stadia wires h I and c c are attached are 
held apart by the hoop-spring, shown in the figure, and are ad- 
justed by the capstan-screws d d. 
16 



236 



LAXD-SUB TETIXG. 




It is evident that, in looking through the telescope at a gradu- 
ated rod, a certain portion 
of the rod will be inter- 
cepted between the stadia 
wires, and that the greater 
the distance at which the 
rod is held, the longer will 
be the space on the rod in- 
tercepted by the stadia 
wires. 

Referring to Art. 287, 
Fig. 201, we see that the 
objective of the telescope forms an image, B, of the arrow, A. 
A may represent the part of the rod intercepted by the stadia 
wires, and B the distance between the wires. The farther the rod 
is carried from the telescope, the nearer the image is formed to 
the objective. If the rod were at an infinite distance, the image 
would be formed at the principal focus of the objective. 

Call the distance of the principal focus from the lens,/; the 
distance from the lens to the rod held at any point, p ; the dis- 
tance from the lens to the image, qj the space intercepted on 
the rod by the stadia wires, h; and the distance apart of the stadia 
wires, a. 

As p increases, £ increases, q decreases, and a remains constant. 
From similar triangles, Fig. 201, we have : 

p : q \\k :a, [1.] 

and from the principles of optics — 

[2.] 



1+ 


1 _ 1 


From 


q a 


From [2] 


P-P , 


. P 


a 


and p : 


={*+/. 



[3.] 



THE STADIA OR TELEMETER. 



237 



Formula [3] is not perfectly accurate, as p and q are measured 
from the surface of the lens instead of its center, and the objective 
of the telescope is not a simple double-convex lens. It is, how- 
ever, sufficiently exact for this purpose. 

We see by the formula [3] that, as / and a are constants, the 
distance, p, from the objective to the rod is equal to the reading 
on the rod, multiplied by a constant quantity, plus the principal 
focal distance of the objective. To obtain the distance from the 
center of the instrument to the rod, it is also necessary to add the 
distance from the center of the instrument to the objective. Call 
this distance c. Then, for the distance from the center of the in- 
strument to the rod, we have : 



f 

distance = — Tc 4- f 4- c. 
a J 



[*•] 



The distance from the objective to the center of the instrument 
is not precisely the same for all lengths of sight. The farther off 
the object sighted to is, the nearer the image will be formed to the 
objective, and hence the objective must be drawn in, in order that 
the image may be formed at the cross-hairs. When the object 
sighted to is near, the image is formed farther from the objective, 
and the objective-slide must be moved out in order that the image 
may be formed at the cross-hairs. Hence, we see that the quantity 
c is not rigidly con- Fig ^ 

stant. The differ- kl /iE 

ence in value, how- 
ever, is not enough 
to be taken into 
consideration. A 
mean value of c can 
be determined by 
sighting to some 
object at a distance 
of the mean length 
of sight (say five 
hundred feet), and 

then measuring the distance from the objective to the center of 
the telescope-axis. 




238 LAND-SUB VEYINO. 

347. Formula [4] is for level ground. For sloping ground, this 
must be modified. In Fig. 250 let A be the center of the telescope- 
axis ; C E, the reading on the rod ; D, the point on the rod where 
the center cross-hair intersects the rod ; A B, the horizontal dis- 
tance ; H, a point in front of the object-glass, and at a distance 
equal to its focal length ; e, the angle of elevation ; M L, perpen- 
dicular to the line of sight ; /, a, c, and Jc as in [4]. Then we 
have : 

M L = C E cos. e = Jc. cos. e and H D = — h cos. e. 

a 
f 
H I = H D cos. e — J -Jc cos. 2 e. 
a 

AB = AN + KB(=HI). 

f 

.-. AB = (c+f) cos. e+*-£cos. 2 e. [5.] 

The height B D = A B tan. e 

B D = (c +/) sin. e + f -k EIL?_!. [6.] 

To find the value of a in any case, measure off from the point 
over which the instrument is set a base-line, B (say one thousand 
feet), and hold the stadia-rod at the farther end. Let the reading 
on the rod be Jc'. 

Then, by [4] B =■£#+/+* 

and a 



B-f-c 
Substituting this value of a in equations [5] and [6], we have : 

Horizontal distance = (c +/) cos. e + t> (B — /— c) cos. 2 e. [7.] 

Difference of level = (c +/) sin. e + — r? (B — / — c) sin. 2 e. [8.] 

Z fc 

348. The Stadia-Tables * given in this volume were calculated 
from formulas [7] and [8], using the following values : 

The measured base, B = 1,000 feet, and Jc' = the reading on 
the rod for that distance — i. e., the distance indicated by the stadia- 
reading is 1,000 feet. 



* Calculated by Alfred Noble and William T. Casgrain, and used on the United 
States Lake Survey. 



THE STADIA OR TELEMETER, 239 

(c+/) = 1-4 feet. 

The quantities in the columns headed a and b are computed 
respectively from the expressions (c-\-f) cos. e, and (c +/) sin. e, 
in the formulas. They are constant for all readings if the angle e 
remains the same. 

The horizontal distances, and the differences of level, are com- 
puted by the tables in a manner similar to that employed in calcu- 
lating latitudes and departures with a table. 

Example. Let e = 4° 27', and h = reading corresponding to 
1,384 feet when the ground is horizontal. 

Take from the table as follows : 

HOEIZONTAL DISTANCE. DIPFEEENCE OF LEVEL. 

For 1,000 992-6 For 1,000 077'2 

" 300 297-78 u 300 23*17 

" 80 79-407 ' " 80 6'180 

" 4 3-9703 u 4 -3090 

" (o + f) cos. e.. 1-3958 " (c + f) sin. e. . -1086 

1375-1531 106-9676 

The difference of level given by formula [8] is the difference 
in height between the instrument at A and the point where the 
central cross-hair strikes the rod at O. The difference between 
the height of the instrument above the ground, and the height 
of C above the ground, must be applied as a correction to the 
difference of level, obtained by the formula, to get the true differ- 
ence of height of the ground at the instrument, and at the rod. 

349. The stadia- wires may be adjusted to use with a rod already 
graduated to feet and decimals, or, if the wires are fixed, a rod may 
be graduated to suit the wires. 

In the first case the wires are adjusted so that one foot is in- 
cluded between the wires at a given distance (50 or 100 feet) plus 
the constant c. Suppose the space included between the wires was 
one foot, at a distance from the center of the instrument of 100 
feet + c. Then, if the reading on the rod held at some unknown 
distance was 3*46 feet, the distance would be 346 feet -f c. 

If the wires are fixed, measure off from the center of the instru- 
ment 500 feet + c, and note the space on the rod, intercepted by 
the cross-hairs at that distance. Divide this space into five equal 



240 LAND-SURVEYING. 

parts, subdivide the parts to tenths and hundredths, and graduate 
the remainder of the rod with similar divisions. This rod can then 
be used in the same way as the rod, graduated to feet, was in the 
first case. Suppose, on holding up this rod at an unknown dis- 
tance, that the stadia- wires intercepted 3 '67 of the parts. Then 
the distance is 367 feet + & 

The rod may be supplied with one or two targets, or may 
be used as a "speaking-rod" — that is, it maybe graduated and 
marked so as to be read by the observer at the instrument. 

For forms of targets, and methods of graduating and marking 
rods, see subject "Bods," Part II. 

350. Several different formulas and methods have been used in 
stadia-surveying, depending upon the object and extent of the 
survey, and the degree of accuracy required. Another method 
is given in the following communication,* together with results 
in practice : 

351. Results of Telemeter Traverse between Triangulation-Points 
on the Shores of Lake George, New York. 

Instrument. Engineer's transit of "W. & L. E. Gurley. Focal length = 
0*565 feet ; distance of cross-wires from center of instrument = 0*13 feet. 
One extra cross-wire was added to the diaphragm. At 103 feet from the 
center of the instrument, the distance included between the wires was found 
to be 1-0253 feet— 

by the formula, t = 01005 d — 0-01 feet, [1.] 

or, d = 99-48 t + 1 foot, [2.] 

where t = distance included between the wires at any distance, d, from the 
center of the instrument. 

Stadia-Rod or Telemeter. This was graduated especially for the instru- 
ment from formula [1], the zero of graduation being displaced 0-01 foot to 
allow for the constant of the formula. The least reading of the rod was 21 
feet. Distances were estimated and recorded to single feet. 

Circumstances of Measurement. Traverse-lines were run between trian- 
gulation-points ; the distances between the latter were computed from the 
traverse and compared with the results from triangulation, in nine cases. 
The aggregate length of these nine lines was about 10^- miles. 

Four closed traverses were run around islands, and the errors of closure 
were obtained. 

The lines of sight generally passed over water, which circumstance was 
favorable to precise reading. 

* From Horace Andrews, C. E., assistant on New York State Survey. 



TEE STADIA OR TELEMETER. 



241 



The results of comparison are given below. They indicate that the con- 
stants used in graduating the telemeter-rod were not exactly obtained. The 
error of measurement averaged + 2*2 feet to 1,000. If this allowance had 
been made in graduating the rod, or this constant error had been allowed 
for, the purely accidental errors would have been only ±1*2 foot to the 
1,000. The law of propagation of errors of length is favorable to close linear 
measurements with the telemeter upon traverse-lines, as was found to be 
actually the case here. In traverse-lines, the larger part of the total error is 
due to angular errors which overweigh the linear ones, unless exceptional 
means are taken to avoid this. 



(1) 


(2) 


(3) 


(4) 


(5) 


(1) = Distances between tri- 
angulation-points as com- 
puted from traverse. 


feet. 
5183-5 


feet. 
+ 12-8 


feet. 
+ 2-47 


feet. 
+ 0-31 


9 


3988-0 


+ 7-5 


+ 1-88 


—0-28 


7 


(2) = Distance by traverse 


4925-7 


+ 7-6 


+ 1-54 


—0-62 


9 


minus distance by tri- 
angulation. 


8427-8 


+ 11-7 


+ 1-39 


-0-77 


17 




2995-0 
3104-6 


+ 15-0 
+ 9-7 


+ 5-01 
+ 3-12 


+ 2-85 
+ 0-96 


7 
5 


(3) = Error to 1,000 feet, in- 
cluding constant error. 


9593-2 


+ 20-2 


+ 2-11 


-0-05 


15 


(4) = Purely accidental error 


6987-9 


+ 6-0 


+ 0-86 


—1-30 


20 


to 1,000 feet. 


9850-0 


+ 10-0 


+ 1-02 


-1-14 


20 


(5) = Number of sides to 
traverse, or number of 
stadia-readings. 


55055-7 




+ 2-16 


±1-21 = V*T 

n 





CLOSED TEAVEESES. 



LOCALITY. 


(i) 


(2) 


(3) 


(4) 


(1) = Sum of distances by 
traverse. 

(2) = Closing error. 

(3) = Number of sides to 
closed traverse. 

(4) = Error to 1,000 feet, in- 
cluding constant error. 


Mother Bunch Islands. 

Vicar's Island 

Harbor Islands 

Hatcher Island 


feet. 
4061 
2316 

5722 
1610 


feet. 

13-9 
7-1 
1-9 
3-5 


feet. 

14 

10 

12 

6 


feet. 
3-42 
3-06 
0-33 
2-17 



352. In 1881 a stadia-survey for a road was made in Mexico,* 
from Culiacan to Durango. Two different routes were followed, 
one in going up the mountains to Durango, and the other on the 
return to Culiacan. The total distance run was 606 miles, and 
difference of elevation 11,000 feet. When the entire traverse was 
closed, the error of closure was found to be 1,100 feet. 



The greater part of the work was done by W. B. Landreth, C. E. 



CHAPTER V. 

OBSTACLES IN ANGULAR SURVEYING. 

353. The obstacles, such as trees, houses, hills, valleys, rivers, 
etc., which prevent the direct alinement or measurement of any 
desired course, can be overcome much more easily and precisely 
with any angular instrument than with the chain, methods for 
using which were explained in Chapter II. They will, however, 
be taken up in the same order. As before, the given and measured 
lines are drawn with fine full lines ; the visual lines with broken 
lines ; and the lines of the result with heavy full lines. Part of 
the demonstrations of the problems are given, and part are left as 
exercises for the student. 

PERPENDICULARS AND PARALLELS. 

354. Erecting Perpendiculars. To erect a perpendicular to a 
line at a given point, set the instrument at the given point, and, if 
it be a compass, direct its sights on the line, and then turn them 
till the new bearing differs 90° from the original one. A conven- 
ient approximation is to file notches in the compass-plate, at the 
90° points, and stretch over them a thread, sighting across which 
will give a perpendicular to the direction of the sights. 

The transit being set as above, note the reading of the vernier, 
and then turn it till the new reading is 90° more or less than the 
former one. 

355. To erect a perpendicular to an inacessible line, at a given 
point of it. Let A B be the line and A the point. Calculate the 
distance from A to any point C, and the angle CAB, by the 



PERPENDICULARS AND PARALLELS. 243 

method of Art. 381. Set the instrument -piQ. 251. 

at 0, ^ight to A, turn an angle = C A B, Ar B 

and measure in the direction thus ob- 
tained a distance C P = C A . cos. CAB. 
PA will be the required perpendicular. 

356. Letting fall Perpendiculars. To 

let fall a perpendicular to a line from a given point. With the 
compass, take the bearing of the given line, and then from the 
given point run a line, with a bearing differ- 
ing 90° from the original bearing, till it 




reaches the given line. 

With the transit, set it at any point of 
the given line, as A, and observe the angle 
between this line and a line thence to the 
given point, P. Then set at P, sight to the 
former position of the instrument, and turn a number of degrees 
equal to what the observed angle at A wanted of 90°. The in- 
strument will then point in the direction of the required per- 
pendicular P B. 

357. To let fall a perpendicular to a line from an inaccessible 
point. Let A B be the line and P the 
point. Measure the angles P A B and 
P B A. Measure A B. The angles A P C 
and B P C are known, being the comple- 
ments of the angles measured. Then 
tan. A P 



is A C = A B . 




tan. APC + tan. B P 0' 



Proof: A = P C . tan. A P ; and C B = P C . tan. B P [Trigo- 
nometry, Art. 4]. 

Hence A : B :: tan. APC : tan. BPO ■; and ■ 

A C : A C + B :: tan. A P : tan. A P + tan. BPO. 

Consequently, since AC + CB = AB, AC = AB. '- . 

tan.APC + tan. BPC 

358. To let fall a perpendicular to an inaccessible line from a 
given point. Let C be the point and A B the line. Calculate the 



2±± LAND-SURVEYING. 




angle C A B by the method of Art. 381. 
Set the instrument at 0, sight to A, 
and turn an angle = 90 — C A B. It 
will then point in the direction of the 
required perpendicular, C E. 



359. Running Parallels. To trace a 
line through a given point parallel to a given line. With the com- 
pass, take the bearing of the given line, and then, from the given 
point, run a line with the same bearing. 

With the transit or theodolite, set it at any convenient point 
of the given line, as A, direct it 
on this line, and note the reading. 

Then turn the vernier till the A \ ~ B 

cross-hairs bisect the given point, \ 

P. Take the instrument to this ___ 

point and sight back to the former P Q 

station, by the lower motion, with- 
out changing the reading. Then move the vernier till the reading 
is the same as it was when the telescope was directed on the given 
line, or 180° different. It will then be directed on P Q, a parallel 
to A B, since equal angles have been measured at A and P. The 
manner of reading them is similar to the method of "Traversing." 

360. To trace a line through a given point parallel to an inac- 
cessible line. Let C be the given point 
Fig. 256. 6 r 

and AB the inaccessible line. Find 

the angle CAB, as in Art. 381. Set 
the instrument at C, direct it to A, 
and then turn it so as to make an angle 
with C A equal to the supplement of 
the angle CAB. It will then point in a direction, C E, par- 
allel to A B. 




OBSTACLES TO ALINEMENT. 



245 



OBSTACLES TO ALINEMENT. 

A. To prolong a Line. 

361. The instrument being set at the farther end of a line and 
directed back to its beginning, the sights of the compass, if that 
be used, will at once give the forward direction of the line. A dis- 
tant point being thus obtained, the compass is taken to it and the 
process repeated. The use of the transit for this purpose has 
been fully explained. 

362. By Perpendiculars. When a tree or house obstructing the 
line is met with, place the instrument 
at a point B of the line, and set off 

there a perpendicular to C ; set off a 

another at to D, a third at D to 
E, making D E = B C, and a fourth 

at E, which last will be in the direction of A B prolonged. If 
perpendiculars can not be conveniently used, let B and D E 
make any equal angles with the line A B, so as to make C D 
parallel to it. 



Fig. 257. 



363. By an Equilateral Triangle. At B turn aside from the line 

at an angle of 60°, and measure 
some convenient distance B 0. At 
C turn 60° in the contrary direc- 
tion, and measure a distance C D 
= BO. Then will D be a point in 
the line A B prolonged. At D turn 

60° from C D prolonged, and the new direction will be in the line 

of A B prolonged. This method 

requires the measurement of one 

angle less than the preceding. 




Fig. 259. 



364. By Triangulation. Let 

A B be the line to be prolonged. 
Choose some station C, whence 
can be seen A, B, and a point beyond the obstacle. Measure A B 




246 LAND-SURVEYING. 

and the angles A and B of the triangle ABC, and thence calculate 
the side AC. Set the instrument at C, and measure the angle 
ACD, CD being any line which will clear the obstacle. Let E be 
the desired point in the lines A B and C D prolonged. Then in 
the triangle ACE will be known the side A C and its including 
angles, whence C E can be calculated. Measure the resulting dis- 
tance on the ground, and its extremity will be the desired point 
E. Set the instrument at E, sight to C, and turn an angle equal 
to the supplement of the angle A E C, and you will have the di- 
rection, E F, of A B prolonged. 

365. When the Line to be prolonged is inaccessible. In this 
case, before the preceding method can be applied, it will be neces- 
sary to determine the lengths of the lines A B and A C, and the 
angle A, by the method given in Art. 381. 

366. To prolong a Line with only an Angular Instrument. This 

may be done when no means 
260- of measuring any distance can 

be obtained. Let A B be the 
\ line to be prolonged. Set the 

'"^ instrument at B and deflect 

angles of 45° in the directions 
C and D. Set at some point, 
C, on one of these lines and 
deflect from C B 45°, and mark the point D where this direction 
intersects the direction BD. Also, at C, deflect 90° from B. 
Then, at D, deflect 90° from D B. The intersections of these last 
directions will fix a point E. At E deflect 135° from E C or 
E D, and a line E F, in the direction of A B, will be obtained and 
may be continued.* 

B. To INTERPOLATE POINTS EST A LlXE.. 

367. The instrument being set at one end of a line and directed 
to the other, intermediate points can be found, etc. If a valley in- 

* This ingenious contrivance is due to Mr. R. Hood, in whose practice, while run- 
ning an air-line for a railroad, the necessity occurred. 




OBSTACLES TO ALINEMENT. 247 

tervenes, the sights of the compass (if the compass-plate be very 
carefully kept level crosswise), or the telescope of the transit, 
answer as substitutes for the plumb-line. 

368. By a Bandom Line. When a wood, hill, or other obstacle 
prevents one end of the line, Z, from being seen from the other, A, 
run a random line A B with the com- 
pass or transit, etc., as nearly in the 
desired direction as can be guessed, till 
you arrive opposite the point Z. Meas- 
ure the error, BZ, at right angles to 

A B, as an offset. Multiply this error by 57 T 3 o, and divide the 
product by the distance A B. The quotient will be the degrees 
and decimal parts of a degree contained in the angle B A Z. Add 
or subtract this angle to or from the bearing or reading with 
which A B was run, according to the side on which the error was, 
and start from A, with this corrected bearing or reading, to run 
another line, which will come out at Z, if no error has been com- 
mitted. 

Example : A random line was run, by compass, with a bearing 
of S. 80° E. At twenty chains distance a point was reached oppo- 
site to the desired point, and ten links distant from it on its right. 
Eequired the correct bearing. 

Ans. By the rule, *' = 0°-2865 = 17'. The correct 
J 2,000 

bearing is therefore S. 80° 17' E. If the transit had been used, its 
reading would have been changed for the new line by the same 
17'. A simple diagram of the case will at once show whether the 
correction is to be added to the original bearing or angle, or sub- 
tracted from it. 

If trigonometrical tables are at hand, the correction will be 

more precisely obtained from this equation : Tan. B A Z = -r-=. 

T{ 7 10 

In this example, T -^ = ^t^t, = * n °5 = tan. 17'. 
r AB 2,000 

The 57° *3 rule, as it is sometimes called, may be variously 

modified. Thus, multiply the error by 86°, and divide by one and 

a half time the distance ; or, to get the correction in minutes, 



248 LAND-SURVEYING. 

multiply by 3,438 and divide by the distance ; or, if the error is 
given in feet and the distance in four-rod chains, multiply the 
former by 52 and divide by the distance, to get the correction in 
minutes. 

The correct line may be run with the bearing of the random 
line by turning the vernier for the correction. 

369. By Latitudes and Departures. When a single line, such as 
A B, can not be run so as to come opposite to the 

Fig. 262. ^ YQn p oint ^ proceed thus with the compass: 

A Eun any number of zigzag courses, A B, B C, CD, 

DZ, in any convenient direction, so as at last to 
arrive at the desired point. Calculate the latitude 
and departure of each of these courses and take 
their algebraic sums. The sum of the latitudes 






AT"" 



ffl'd \ ^^ ^ e e( l Iial to A X, and that of the departures to 

fSJJ XZ. Then is tan. Z A X = ^| ; i. e., the alge- 

braic sum of the departures divided by the alge- 
p braic sum of the latitudes is equal to the tangent 

of the bearing.* 

370. When the transit is used, any line may be taken as a 
meridian — i. e., as the line to which the following lines are re- 
ferred ; as in "Traversing," Art. 335, all the successive lines were 
referred to the first line. In Fig. 263 the same lines as in the 
preceding figure are represented, but they are referred to the 
first course, A B, instead of to the magnetic meridian as before, 
and their latitudes are measured along its produced line, and its 
departures perpendicular to it. As before, a right-angled triangle 
will be formed, and the angle ZAY will be the angle at A be- 
tween the first line AB and the desired line AZ. 

This method of operation has many useful applications, such as 
in obtaining data for running railroad-curves, etc., and the student 
should master it thoroughly. 

* The length of the line A Z can also be at once obtained, since it is equal to the 
square root of the sum of the squares of AX and XZ, or to the latitude divided by 
the cosine of the bearing. 



OBSTACLES TO MEASUREMENT. 



249 




The desired angle (and at the same time 
the distance) can be obtained, approximately, 
in this and the preceding case, by finding in a 
traverse-table the final latitude and departure 
of the desired line (or a latitude and departure 
having the same ratio), and the bearing and 
distance corresponding to these will be the 
angle and distance desired. 

371. By Similar Triangles. Through A measure any line C D. 

Take a point E, on the line CB, 
beyond the obstacle, and from it 
set off a parallel to CD, to some 
point, F, in the line D B. Meas- 
ure EF, CD, and C A. Then 
this proportion, C D : C A : : E F : 
E G-, will give the distance E Gr, 
from E to a point in the line A B. So for other points. 




Fig. 265. 



372. By Triangulation. When obstacles prevent the preceding 
methods being used, if a point, C, can be found from which A and 
B are accessible, measure the distances 
C A, C B, and the angle A C B, and 
thence calculate the angle CAB. 
Then observe any angle A C D beyond 
the obstacle. In the triangle A C D 
a side and its including angles are 
known to find C D. Measure it, and 
a point, D, in the desired line will be obtained. 




OBSTACLES TO MEASUREMENT. 

A. When Both Ends of the Line aee accessible. 

373. The methods given in the preceding articles for prolong- 
ing a line and for interpolating points in it will generally give the 
length of the line by the same operation. The method of latitudes 
and departures is very generally applicable. So is the following. 



250 



LAXD-SUR VEYIXG. 



Fig. 266. 



A v- 




374. By Triangulation. Let AB 

be the inaccessible distance. From 
any point, 0, from which both A 
and B are accessible, measure CA, 
C B, and the angle A C B. Then in 
the triangle ABO two sides and the 
included angle are known to find the 
side A B. * 



375. By Angles to Known Points. The length of a line, both 
ends of which are accessible, may also be determined by angles 
measured at its extremities between it and the directions of two or 
more known points. But, as the methods of calculation involve 
subsequent problems, they will be postponed. 



B. When One End of the Line is inaccessible. 
376. By Perpendiculars. Many of the methods given for the 
chain may be still more advantageously employed with angular 
instruments, which can so much more easily and precisely set off 
the perpendiculars. 



377. By Equal Angles. Let AB be the inaccessible line. 
set off A C perpendicular to A B, 
and as nearly equal to it, by estima- 
tion, as the ground will permit. At " ^ — 
measure the angle A C B, and turn 
the sights or vernier till ACD = 
A C B. Find the point, D, at the 
intersection of the lines C D and 
B A produced. Then is A D = A B. 



At A 



Fig. 267. 






378. By Triangulation. Measure a distance A C, about equal 

to A B. Measure the angles at A and C. Then, in the triangle 

ABC, two angles and the included side are known, to find another 

. , . t, AC sin. A B 

side, A B = — = A -p n — . 

sin. ABC 

* In this figure and the following ones the angular point inclosed in a circle indi- 
cates the place at which the instrument is set. 



OBSTACLES TO MEASUREMENT. 



251 



When the compass is used, the angles be- 
tween the lines will be deduced from their re- 
spective bearings. 

If the angle at A is 90°, A B = A C . tang. 
ACB. 

If the angle A C B = 45°, then A 0= A B ; 
but this position could not easily be obtained, 
except by the use of the sextant, a reflecting 
instrument, described in Part V. 



Fig. 268. 




>B 



Fig. 268'. 



379. When One Point can not be seen from the other. Choose 
two points, and D, in the line of A, 
and such that from 0, A, and B can be 
seen, and from D, A, and B. Measure 
AC, AD, and the angles C and D. 
Then, in the triangle B C D, are known 
two angles and the included side, to 
find CB. Then, in the triangle A B C, 
are known two sides and the included 
angle, to find the third side, A B. 




380. To find the Distance from a Given Point to an Inacces- 
sible Line. In Fig. 254, Art. 358, the required distance is C E. 
The operations therein directed give the line C A and the angle 
C A B, or C A E. The required 
distance C E = C A . sin. C A E. 



Fig. 269. 



C. When Both Ends of the Line 

AEE INACCESSIBLE. 

381. General Method. Let 
A B be the inaccessible line. 
Measure any convenient distance, 
CD, and the angles A C D, 
BCD, ADC, BDC. 

Then, in the triangle C D A, 
two angles and the included side 
are given, to find C A. In the 
17 



>E 




:he izicl-iei 


:: triangles. 


ABC. 


Astazces ~i:h 



252 LASD-SUR YEYIXG 

:__. ngie C D B. :~c> azgies and the incinded side are given. :: h::z 
C R. Then, in the triangie ABC, ;~o sides and " 
air:, are riven. :: rind A B, 

and landing AB from the rriangle ABD instead :: 

r — -.._.- r- .„-v.v-..-. -r _*in. ADC . sin- CBI 

front tie tArizzA— 

Tan. i CAB — A3 O = tan, ^! c — A , c:t. jAC: 
TneaA CAB = *'CAB — ABC - T 3AB + ACB) 

... s m. B D C . em. A : E 

l-„v. a b _ LU^ CBDiS ^ CAB - 

Li- : - ' ;:::i:-::>AZ: ierignzte the azAes as A. 3. C : ant 

the sides otz isite t: then as .:. :. ■:. Let C A = .:. Ize traznAe 3 CI 1 ra~es 

_ , , "^ T . r .sin.BDC _ . . - . -. - . . 

i::z,. Art. _a. .zeirezz F. a = a . — _.— _ . ^ze trtanz.e A . _ sazzt- 

szz. I E - 

"•-•*'■«■ -"ET~ 

In the t. AzAe A B C. — e have TrA.. Art. 12. ihe::eza IT, 
tan. t A — L : : : : r z :: t — : : : — : ; 

whence tan . - A-h = —A . c::. t C. [1.] 

_ . ... / _ 1 

Let X be an .: zziz; .r~ anzze. sziz :_:.: : = .:: , tat _ ~_ez: 7 tan. ^ = — . 

: 

Z iviauig the sezzna nzezzbe: : t e _z:zi:z Ah at t~e :zz ' : eA ~ '17 .:. azi. szb- 

^ . : . .. _. 1 — taz.X 

str.ntzz:; tan. A tor — . we ret tan. - A — - = — . cot. 1 C. 

.: " 1 - tan ^ 

^mte tan. -=r " = _. — e may stit stttzte it trr - zz zze t rezeoizz e 1 z 



1 r* t»\ at. -r: — tz, a _ 

azt • ■ e .z: tan 7 A — r> = : — ■ . cot. t L . 

tan. 4S " — tzz . _ 

I: :z: the ezare^si : z :" r the tzz rent :t the ::ztetez:e :: t~ : a:is ArA.. 

Art. S^. tze zr^iAzzz zrzttiin redntes t: taz A : — A aza tie ::z::z 
becizzes 

tan. i A-S = taz. ^o : -X . :::. : C, [2.] 

In the ez/zztizz tan. X = — . sztstitzte tze ~Azes :: " :zb 






This rivea 



►-.- A = - 5 "- a: : _ - ; - ~- -" _ = -_-'-• I ? . stn. C B D 
s:z CAB ' 5;z C I I ~ rA. C A D . sin. B I 

A — 3 : .s then tbtaine-a :-v e:nzrA"n AT A-h is the szzt Anient 0: 
C ; therefore, the anzA A is hz: -■ z. 



OBSTACLES TO MEASUREMENT 



253 



Then 



AB = 



a sin. C d . sin. BDC. sin. ACB 



sin. A sin. CBD. sin. CAB 

The use of the auxiliary angle K avoids the calculation of the sides a and b. 

Example. Let D — 7,106*25 feet ; A D = 95° 17' 20" ; B D = 61° 
41' 50"; ADC = 39° 38' 40"; BD C = 78° 35' 10" ; required A B. 

The figure is constructed with these data on a scale of 5,000 feet to 1 
inch = 1 : 60000. 

By the above formulas, K is found to be 30° 26' 5"; CAB = 113°55' 
37" ; and, lastly, A B = 6598'32. 

Both the methods may be used as mutual checks in any important case. 

If the lines A B and C D crossed each 
other, as in Fig. 270, instead of being situ- 
ated as in the preceding figure, the same 
method of calculation would apply. 

->B 



Fig. 270. 



382. Problem. To measure an inac- 
cessible distance^ A B, when a point, C, in 
its line can be obtained. Set the instru- 
ment at a point, D, from which A, B, and 
O can be seen, and measure the angles 

CD A and A D B. Measure also the line D C and the angle C. Then in 

the triangle A C D two angles and 




Fig. 271. 




the included side are given to find 
A D. In the triangle DAB, the an- 
gle D A B is known (being equal to 
A C D + C DA), and A D having 
been found, we again have two an- 
gles and the included side to find 
AB. 



383. Problem. To measure an 
inaccessible distance, A B, when only 
one point, C, can be found from 

which both ends of the line can be seen. Consider C A and C B as distances 

to be determined, having one end ac- 
cessible. Determine them as in Art. 

378, by choosing a point D, from which 

C and A are visible, and a point E, from 

which C and B are visible. At C ob- 
serve the angles D C A, A C B, and B C E. 

Measure the distances C D and C E. 

Observe the angles ADC and B E C. 

Then in the triangle ADC, two angles 

and the included side are given, to find 

C A ; and the same in the triangle C B E, 

to find C B. Lastly, in the triangle ACB two sides and the included angle 

are known, to find A B. 



Fig. 272. 




254 LAXD-SUR VEYISG. 

384. Problem. To measure an inacessible distance, A B, when no point 
can be found from which the two ends can be seen. Let C be a point from 
which A is visible, and D a point from which B is visible, and also C. 

Measure C D. Find the distances C A 

Fig. 273. an d D B, as in the preceding problem, 

i\ ~ ~~ ^ i. e., choose a point E, from which A 

^f^^^^Z^^Z^Zl /^r anc ^ ® are visible, and another point, F, 

from which D and B are visible. Meas- 




V\-'" ■ ■■ . nre EC and D F. Observe the angles 

" / i A EC, EC A, BDF, and DFB; and 

' —-""I! at the same time the angles A C D and 



C D B, for the subsequent work. Then 
C A and D B will be found, as were C A 
and C B in the last problem. Then in the triangle C D B, two sides and the 
included angle are known to find C B and the angle D CB ; and, lastly, in the 
triangle A C B, two sides and the included angle (the difference of A C D 
andDCB) to find A B. 

385. Problem. Given the angles observed, at the ends of a line which 
can not be measured, between it and the ends of a line of 'known length but 
inaccessible, required the length of the former line. This problem is the con- 
verse of that given in Art. 381. Its figure, 269, may represent the case, if 
the distance AB be regarded as known and CD as that to be found. Use 
the first and second formulas as before, and invert the last formula, obtaining 
CD- AB siD - CBD. sin. CAB 

~~ sin. BDC . sin. ACB' 

This problem may also be solved, indirectly, by assuming any length for 
C D, and thence calculating, as in the first part 
of Art. 381, the length of A B on this hypothesis. Fig. 274. 

The imaginary figure thus calculated is similar to . B'_ 

the true one ; and the true length of C D will be j\ / /' \ 

given by this proportion: Calculated length of / pS x' 

A B : true length of A B : : assumed length of Q' 1 / \' 
C D : true length of C D. ">0\ I 

The length of CD can also be obtained graph- ^ -/ ^ ^- i 

ically. Take a line of any length, as CD', and "--^ 

from C and D 7 lay off angles equal to those ob- "^^ \ J 

served at C and D, and thus fix points A, B'. ""*&») 

Produce A B' till it equals the given distance A B, 

on any desired scale. From B draw a parallel to B'D', meeting AD' pro- 
duced in D ; and from D draw a parallel to D' C meeting AC produced in 
C. Then C D will be the required distance to the same scale as A B. 

386. Problem. Three points, AB C, being given by their distances from 
each other, and two other points, P and Q, being so situated that from each of 
them two of the three points can be seen and the angles A P Q, B P Q. C Q P, 
B Q P, be measured, it is required to determine the positions or P and Q. 



OBSTACLES TO MEASUREMENT. 255 

Construction. Begin by describing a circle passing through A and B, 
and having the central angle subtended by A B, equal to twice the given 
angle A P B, and thus containing 
that angle. The point P will lie Fig. 275. 

somewhere in its circumference. „ -B^-~ ;~-^ 

Describe another circle passing y'' y^^**^>^ 

through B and 0, and having a / 'y*// n ^n^\. ^ 

central angle subtended by B /y^^ / / \ s \ ^^^\ q 

equal to twice the given angle \ ~"~-y'-^J^ i--""^^ \ 

BQO. The point Q will lie \/ _ V^^C^j ---$Q 

somewhere in its circumference. *'w "*\ / ,' 

From A draw a line making 

with AB an angle = B P Q, and \^ x y' 

meeting at X the circle first "--—_— ---"'.-- „, -' 

drawn. From C draw a line mak- 
ing with C B an angle = B Q P, and meeting the second circle in Y. Join 
X Y and produce it till it cuts the circles in points P and Q, which will 
be those required ; since BPX = BAX = BPQ; and BQY = BCY = 
BQP. 

Calculation. In the triangle ABC, the sides being given, the angle 
A B C is known. In the triangle A B X, a side and all the angles are known, 
to find B X. In the triangle C B Y, B Y is similarly found. By subtracting 
the angle ABC from the sum of the angles A B X and C B Y, the angle 
X B Y can be obtained. Then in the triangle X B Y, the sides B X, BY, 
and the included angle are given to find the other angles. Then in the tri- 
angle B P X are known all the angles and the side B X to find B P. In the 
triangle B Q Y, B Q is found in like manner. Finally, in the triangle B P Q, 
P Q can then be found. 

If desired, we can also obtain A P in the triangle A P B ; and C Q in the 
triangle C B Q. 

387. Problem. Four points, A, B, C, D, being given in position, by 
their mutual distances and directions, and two other points, P and Q, being 
so situated that from each of them two of the four points can be seen and the 
angles A P B, A P Q, P Q C, and P Q D measured, it is required to determine 
the position of P and Q. 

Consteuction. Begin, as in the last article, by describing on AB the 
segment of a circle to contain an angle equal to APB. From B draw a 
chord B E, making an angle with B A equal to the supplement of the angle 
A P Q. On CD describe another segment to contain an angle equal to 
C QD. From C draw a chord CF, making an angle with C D equal to the 
supplement of the angle D QP. Draw the line EF, and it will cut the two 
circles in the required points P and Q. 

For, the angle APQ in the figure equals the measured angle APQ, be- 
cause the supplement of the former, EPA, equals the supplement of the lat- 
ter, since it is measured by the same arc as the angle ABE, equal to that 
supplement by construction. So too with the angle D QP. 



256 



LAND-SUR VEYIXG. 



Calculation. To obtain P Q = E F — EP — QF, we proceed to find 
those three lines thus : In the triangle ABE, we know the side A B, the 
angle ABE, and the angle A E B = A P B ; whence to find E B. In the 
same way, the triangle C F D gives F 0. In the triangle E B C are known 

Fig. 276. 




E B and B 0, and the angle EBC=ABC— A BE; whence E C and the 
angle E B are fonnd. In the triangle EOF are known EC, F C, and the 
angle ECF=BCD — ECB — FCD; whence we find E F, and the angles 
CEFandCFE. 

In the triangle B E P, we have E B, the angle BEP = BEC + CEP, 
and the angle BPE = BPA + APE; to find E P and P B. In the triangle 
Q C F, we have C F, and the angles C Q F and C F Q, to find Q C and Q F. 
Then we know PQ = EF — EP-QF. 

The other distances, if desired, can be easily found from the above datn, 
some of the calculations, not needed for P Q, being made with reference to 
them. In the triangle ABP, we know A B, BP, and the angle BAP, to 
find the angle ABP and A P. In the triangle Q D C we know Q C, C D, 
and the angle C Q D, to find the angle QCD and Q D. In the triangle 
P B C, we know P B, B C, and the angle P B C = A B C — A B P, to find 
P C. Lastly, in the triangle Q C B, we know Q C, C B, and the angle Q C B 
= D C B - D C Q, to find Q B. 

The solution of this problem includes the two preceding ; for, let the line 
B C be reduced to a point so that its two ends come together and the three 
lines become two, and we have the problem of Art. 386 ; and let the line 
A B be reduced to a point, B, and C D to a point, C, and we have but one 
line, and the problem becomes that of Art. 3S5. 

In these three problems, if the two stations lie in a right line with one of 
the given points, the problem is indeterminate. 



388. Problem of the Eight Points. Four points, A. B. C. D, are inac- 
cessible, but visible from four other points, E, F, G, H ; it is required tofnd 
the relative distances of these eight points ; the only data being the obserra- 



TO SUPPLY OMISSIONS. 



257 



tion, from each of the points of the second system, of the angles under which 
are seen the points of the first system. 

This problem can be solved, F IGt 277. 

but the great length and com- 

plication of the investigation S * 

and resulting formulas render it • D 

more a matter of curiosity than y. * 

of utility. It maybe found in 
Puissant's " Topographie," page 
55 ; Lefevre's " Trigonometrie," 
page 90, and Lefevre's " Arpen- 
tage," No. 387. 




TO SUPPLY OMISSIONS. 

389. Any two omissions 
in a closed survey, whether 
of the direction or of the 

length, or of both, of one or more of the sides bounding the area 
surveyed, can always be supplied by a suitable application of the 
principle of latitudes and departures, although this means should 
be resorted to only in cases of absolute necessity, since any omis- 
sion renders it impossible to "test the survey." In the following 

articles the survey will be considered 
to have been made with the compass. 
All the rules will, however, apply to 
a transit survey, the angles being re- 
ferred to any line as a meridian, as in 
"traversing." 

To save unnecessary labor, the ex- 
amples in the various cases now to 
be examined will all be taken from 
the same survey, a plat of which is 
given in the margin on the scale of 40 chains to 1 inch (1 : 31,680), 
and the field-notes of which, with the latitudes and departures 
carried out to five decimal places, are given on page 258.* 

* The teacher can make any number of examples for his own use by taking a 
tolerably accurate survey, striking out the bearing and distance of any one course, 
and calculating it precisely as in Case 1, given below. He can then omit any two 
quantities at will, to be supplied by the student by means of the rules now to be 
given. 




258 



LAND-STIR Y EYING. 



. EC 

< £ 


BEARINGS. 


gg 


LATITUDES. 


DEPARTURES. 


N. 


s. 


E. 


w. 


A 
B 

D 
E 
F 


North. 
N. 32° E. 
N. 80° E. 
S. 48° E. 
S. 18° W. 
N. 73° 28' 21" W. 


1284 
1782 
2400 
2700 
2860 
46211 


1284-00000 

1511-22171 

416-75568 

1314-69682 


1806-65262 
2720-02159 




944-31619 
2363-53872 
2006-49096 




883-78862 
4430-55725 


4526-67421 4526-67421 5314-34587 

1 1 


5314-34587 



Case 1. When the length and the bearing of any one side are 



390. Find the latitudes and the departures of the remaining 
sides. The difference of the north and south latitudes of these 
lines is the latitude of the omitted line, and the difference of their 
departures is its departure. This latitude and departure are two 
sides of a right-angled triangle of which the omitted line is the 
hypotenuse. Its length is therefore equal to the square root of 
the sum of their squares, and the quotient of the departure divided 
by the latitude is the tangent of its bearing. 

In the above survey, suppose the course from F to A to have 
been omitted or lost. The difference of the latitudes of the re- 
maining courses will be found to be 1314*69682, and the difference 
of the departures to be 4430-55725. The square root of the sum 
of their squares is 4621*5 ; and the quotient of the departure 
divided by the latitude is the tangent of 73° 28' 21". The defi- 
ciencies were in north latitude and west departure, and the omitted 
course is therefore N". 73° 28' 21" W., 4621-5. 

Case 2. 1Yhe?i the length of one side and the hearing of an- 
other are wanting. 



391. When the Deficient Sides adjoin Each Other. Find, as in 
Case 1, the length and bearing of the line joining the ends of the 
remaining courses. This line and the deficient lines will form a 
triangle, in which two sides will be known, and the angle between 
the calculated side and the side whose bearing is given can be 



TO SUPPLY OMISSIONS. 



259 



found. The parts wanting can then be obtained by the common 
rules of trigonometry. 

In the figure, let the length of E F and the bearing of F A be 
the omitted parts. The difference of the sums of the 1ST. and S. 
latitudes, and the E. and W. depart- 
ures of the complete courses from A 
to E, are respectively 1405 -32477 north 
latitude, and 5314*34587 east depart- 
ure. The course, E A, corresponding 
to this deficiency, we find, by proceed- 
ing as in Case 1, to be S. 75° 11' 15" 
W., 5497-026. The angle AEF is 
therefore = 75° 11' 15" - 18° = 57° 
11' 15". Then in the triangle AEF 

are given the sides A E, A F, and the angle A E F to find the re- 
maining parts, viz., the angle AFE = 91° 28' 21", whence the 
bearing of F A = 91° 28' 21" - 18° = N. 73° 28' 21" W. ; and the 
side EF = 28-60. 




392. When the Deficient Sides are separated from Each Other. 

A modification of the preceding method will still apply. In this 
figure let the omissions be the bearing of F A and the length of 

C D. Imagine the courses to change 
places without changing bearings or 
lengths, so as to bring the deficient 
lines next to each other by transfer- 
ring CD to AG, AB to GH, and BO 
to HD. This will not affect their 
latitudes or departures. Join GF. 
Then in the figure D E F G H the lati- 
tudes and departures of all the sides 
but F G are known, whence its length 
and bearing can be found as in Case 1. Then the triangle AGF 
may be treated like the triangle A E F in the last article, to obtain 
the length of A G = C D, and the bearing of F A. 

Otherwise, by changing the meridian. Imagine the field to 
turn around till the side of which the distance is unknown be- 




260 



LAND-SUR YEYING. 



comes the meridian — i. e., comes to be due north and south — all 
the other sides retaining their relative positions, and continuing to 
make the same angles with each other. Change their bearings ac- 
cordingly. Find the latitudes and departures of the sides in their 
new positions. Since the side whose length was unknown has been 
made the meridian, it has no departure, whatever may be its un- 
known length ; and the difference of the columns of departure will 
therefore be the departure of the side whose bearing is unknown. 
The length of this side is given. It is the hypotenuse of a right- 
angled triangle, of which the departure is one side. Hence the 
other side, which is the latitude, can be at once found, and also the 
unknown bearing. 

Put this latitude in the table in the blank where it belongs. 
Then add up the columns of latitude, and the difference of their 
sums will be the unknown length of the side which had been made 
a meridian.* 

Let the omitted quantities be, as in the last article, the length 
of CD and the bearing of FA. Make CD the meridian. The 

changed bearings can then be 
found to be as in the margin. 
To aid the imagination, turn 
the book around till C D 
points up and down, as north 
lines are usually placed on a 
map. Then obtain the lati- 
tudes of the courses with 
their new bearings and old distances, and proceed as has been 
directed. 



STATIONS. 


OLD BEARINGS. 


NEW BEARINGS. 


A 


North. 


N". 80° W. 


B 


N. 32° E. 


N. 48° W. 


C 


N. 80° E. 


North. 


I) 


S. 48° E. 


N. 52° E. 


E 


S. 18° W. 


S. 62° E. 


F 







Case 3. When the lengths of two sides are wanting. 

393. "When the Deficient Sides adjoin Each Other. Find the 
latitudes and departures of the other courses, and then, by Case 1, 
find the length and bearing of the line joining the extremities of 
the deficient courses. Then, in the triangle thus formed, are 



* This conception of thus changing the bearings is stated to be due to Professor 
Robert Patterson, of Philadelphia, by whom it was communicated to Mr. John Gum- 
mere, and published by him, in 1814, in his "Treatise on Surveying." 



TO SUPPLY OMISSIONS. 261 

known one side and all the angles (deduced from the bearings) to 
find the lengths of the other two sides. 

Thus, in Fig. 279, let E F and F A be the sides whose lengths 
are unknown. E A is then to be calculated, and its length will be 
found to be 5497*026, and its bearing S. 75° 11' 15" W., whence 
the angle A E F = 75° 11' 15" - 18° = 57° 11' 15" ; A F E = 18° 
+ 73° 28' 21" = 91° 28' 21" ; and E A F = 31° 20' 24" ; whence 
can be obtained EF = 28-60 and FA = 46*215. 

394. When the Deficient Sides are separated from Each Other. 

Let the lengths of B C and D E be those omitted. Again imagine 
the courses to change places, so as to 
bring the deficient lines together, D E 
being transferred to C G, and C D to 
G E. Join B G. Then in the figure 
A B G E F A are known the latitudes 
and departures of all the courses ex- 
cept B G, whence its length and bear- 
ing can be found, as in Case 1. Then 
in the triangle B C G, the angle C B G 
can be found from the bearings of 

C B and B G, and the angle C G B from the bearings of B G and 
G 0. Then all the angles of the triangle are known and one side, 
B G, whence to find the required sides, B C = 1,782, and C G = 
DE = 2,700. 

Otherwise, by changing the meridian. Imagine the field to 
turn around till one of the sides whose length is wanting becomes 
a meridian or due north and south. Change all the bearings cor- 
respondingly. Find the latitudes and departures of the changed 
courses. The difference of the columns of departure will be the 
departure of the second course of unknown length, since the course 
made meridian has now no departure. The new bearing of this 
second course being given in the right-angled triangle formed by 
this course as an hypotenuse, and its departure and latitude, we 
know one side, the departure, and the acute angles, which are the 
bearing and its complement. The length of the course is then 
readily calculated, and also its latitude. This latitude being in- 




262 



LAXD-SUR VEYING. 



STATIONS. 


OLD BEARINGS. 


NEW BEARINGS. 


A 


North. 


N. 32° W. 


B 


N. 32° E. 


North. 





N. 80° E. 


N. 48° E, 


D 


S. 48° E. 


S. 80° E. 


E 


S. 18° W. 


S. 14° E. 


F 


N. 73° 28' 21" W. 


S. 74° 31' 59" W. 



serted in its proper place, the difference of the columns of latitude 
will be the length of that wanting side which had been made a 
meridian. 

Thus, let the lengths of B C and D E be wanting, as in the pre- 
ceding example. 
Make B C a me- 
ridian. The other 
bearings are then 
changed as in the 
margin. Calculate 
new latitudes and 
departures. The 

difference of the departures will be the departure of D E, since 
B C, being a meridian, has no departure. Hence the length and 
latitude of D E are readily obtained. This latitude being put in 
the table, and the columns of latitude then added" up, their differ- 
ence will be the length of B C. 

Case 4. When the hearings of two sides are wanting. 

395. "When the Deficient Sides adjoin Each Other. Find the 
latitudes and departures of the other sides, and then, as in Case 1, 
find the length and bearing of the line joining the extremities of 
the deficient sides. Then, in the triangle thus formed, we have 
the three sides to find the angles and thence the bearings. 



Fig. 2S2. 



396. "When the Deficient Sides are separated from Each Other. 
Change the places of the sides so as to 
bring the deficient ones next to each 
other. Thus, in the figure, supposing 
the bearings of C D and E F to be 
wanting, transfer E F to D G, and 
DE to GF. Then calculate, as in 
Case 1, the length and bearing of the 
line joining the extremities of the de- 
ficient sides, C G in the figure. This 
line and the deficient sides form a tri- 
angle in which the three sides are given to determine the angles 
and thence the required bearings. 




CHAPTER VI. 

LAYING OUT, PARTING OFF, AND DIVIDING UP LAND. 

LAYING OUT LAND. 

397. Its Nature. This operation is precisely the reverse of those 
of surveying properly so called. The latter measures certain lines 
as they are ; the former marks them out in the ground where they 
are required to be, in order to satisfy certain conditions. The 
same instruments, however, are used as in surveying. 

Perpendiculars and parallels are the lines most often employed. 
Part of the demonstrations of the problems are left as exercises for 
the student. 

398. To lay out Squares. Eeduce the desired content to square 
chains, and extract its square root. This will be the length of the 
required side, which is to be set out by one of the methods indi- 
cated in the preceding article. 

An acre, laid out in the form of a square, is frequently desired 
by farmers. Its side must be made 316J links of a Gunter's chain ; 
or 208 T Vo feet ; or 69 T 5 ¥ V yards. It is often taken at 70 paces. 

The number of plants, hills of corn, loads of manure, etc., 
which an acre will contain at any uniform distance apart, can be at 
once found by dividing 209 by this distance in feet, and multiply- 
ing the quotient by itself, or by dividing 43,560 by the square of 
the distance in feet. Thus, at 3 feet apart, an acre would contain 
4,840 plants, etc. ; at 10 feet apart, 436 ; at a rod apart, 160 ; and 
so on. If the distances apart be unequal, divide 43,560 by the 
product of these distances in feet ; thus, if the plants were in rows 
6 feet apart, and the plants in the rows were 3 feet apart, 2,420 of 
them would grow on one acre. 



264 LAND-SURVEYING. 

399. To lay out Rectangles. The content and length 
given, both as measured by the same unit, divide the former by 
the latter, and the quotient will be the required breadth. Thus, 
1 acre or 10 square chains, if 5 chains long, must be 2 chains wide. 

The content being given and the length to he a certain number 
of times the breadth. Divide the content in square chains, etc., by 
the ratio of the length to the breadth, and the square root of the 
quotient will be the shorter side desired, whence the longer side 
is also known. Thus, let it be required to lay out 30 acres in the 
form of a rectangle 3 times as long as broad ; 30 acres = 300 
square chains. The desired rectangle will contain 3 squares, each 
of 100 square chains, having sides of 10 chains. The rectangle 
will therefore be 10 chains wide and 30 long. 

An acre laid out in a rectangle twice as long as broad will be 
224 links by 448 links, nearly ; or, 147-J- feet by 295 feet ; or, 49^ 
yards by 98| yards. Fifty paces by one hundred is often used as 
an approximation, easy to be remembered. 

The content being given, and the difference between the length 
and breadth. Let c represent this content, and d this difference. 
Then the longer side = £ d -f- ^ ^/ (d? + 4 c). 

Example. Let the content be 6*4 acres, and the difference 12 
chains. Then the sides of the rectangle will be respectively 16 
chains and 4 chains. 

The content being given, and the sum of the length and breadth. 
Let c represent this content, and s this sum. Then the longer 
side = \s +| *J (s 3 — 4c). 

Example. Let the content be 6*4 acres, and the sum 20 chains. 
The above formula gives the sides of the rectangle 16 chains and 4 
chains as before. 

400. To lay out Triangles. The content and the base being 
given, divide the former by half the latter to get the height. At 
any point of the base erect a perpendicular of the length thus ob- 
tained, and it will be the vertex of the required triangle. 

The content being given and the base having to be m times the 
height, the height will equal the square root of the quotient ob- 
tained by dividing twice the given area by m. 




LAYING OUT LAND. 265 

The content being given and the triangle to be equilateral, take 
the square root of the content and multiply it by 1*520. The 
product will be the length of the side required. This rule makes 
the sides of an equilateral triangle containing one acre to be 480J 
links. A quarter of an acre laid out in the same form would have 
each side 240 links long. An equilateral triangle is very easily set 
out on the ground, as directed under " Platting/' using a rope or 
chain for compasses. 

The content and base being given, and one side having to make 

a given angle, as B, with the base A B, 

2 X A B FlG< 283, 

the length of the side B C = -j-^ = — ^* 

to A B . sin. B 

Example. Eighty acres are to be laid 
out in the form of a triangle, on a base, 
AB, of sixty chains, bearing N. 80° W., 
the bearing of the side B being K 70° 
E. Here the angle B is found from the bearings (reversing one of 
them) to be 30°. Hence B C = 53 '33. The figure is on a scale of 
50 chains to 1 inch = 1 : 39600. 

Any right-line figure may be laid out by analogous methods. 

401. To lay out Circles. Multiply the given content by 7, di- 
vide the product by 22, and take the square root of the quotient. 
This will give the radius, with which the circle can be described 
on the ground with a rope or chain. A circle containing one acre 
has a radius of 178 J links. A circle containing a quarter of an 
acre will have a radius of 89 links. 

402. Town-Lots. House-lots in cities are usually laid off as 
rectangles of 25 feet front and 100 feet depth, variously combined 
in blocks. Part of New York is laid out in blocks 200 feet by 800, 
each containing 64 lots, and separated by streets, 60 feet wide, 
running along their long sides, and avenues, 100 feet wide, on 
their short sides. The eight lots on each short side of the block 
front on the avenues, and the remaining forty-eight lots front on 
the streets. Such a block covers almost precisely 3f acres, and 17-J- 
such lots about make an acre. But, allowing for the streets, land 



266 



LAXD-SURYETIXG. 



laid out into lots, 25 by 100, arranged as above, would contain only 
11*9, or not quite 12 lots per acre. 

Lots in small towns and villages are laid out of greater size and 
less uniformity : 50 feet by 100 is a frequent size for new villages, 
the blocks being 200 feet by 500, each therefore containing 20 lots. 

403. Land sold for Taxes, A case occurring in the State of 
Xew York will serve as an application of the modes of laying out 
squares and rectangles. Land on which taxes are unpaid is sold at 
auction to the lowest bidder — i. e., to him who will accept the 
smallest portion of it in return for paying the taxes on the whole. 
The lot in question was originally the east half of the square lot 
ABCD, containing 500 acres. At a sale for taxes in 1830, TO 
acres were bid off, and this area was set off to the purchaser in a 
square lot, from the northeast corner. Eequired the side of the 

_ ._. square in links. Again, in 

xlG. 284. 

183-4, 29 acres more were thus 
sold, to be set off in a strip of 
equal width around the square 
previously sold. Eequired the 
width of this strip. Once 
more : in 1839, 42 acres more 
were sold, to be set off around 
the preceding piece. Eequired 
the dimensions of this third 
portion. The answer can be 
proved by calculating if the 
dimensions of the remaining 

rectangle will give the content which it should have, viz., 250 — 

(70+29 + 42) =109 acres. 

The figure is on a scale of 40 chains to 1 inch = 1 : 31680. 

404. New Countries. The operations of laying out land for the 
purposes of settlers are required on a large scale in new countries, 
in combination with their survey. There is great difficulty in 
uniting the necessary precision, rapidity, and cheapness. " Tri- 
angular surveying" will insure the first of these qualities, but is 
deficient in the last two, and leaves the laving out of lots to be 



PARTING OFF LAND. 267 

subsequently executed. " Compass-surveying " possesses the last 
two qualities, but not the first. The United States system for 
surveying and laying out the public lands admirably combines an 
accurate determination of standard lines (meridians and parallels) 
with a cheap and rapid subdivision by compass. The subject is so 
important and extensive that it will be explained by itself. 

PARTING OFF LAND. 

405. It is often required to part off from a field, or from an 
indefinite space, a certain number of acres by a fence or other 
boundary-line, which is also required to run in a particular direc- 
tion, to start from a certain point, or to fulfill some other condition. 
The various cases most likely to occur will be here arranged accord- 
ing to these conditions. Both graphical and numerical methods 
will generally be given.* 

The given content is always supposed to be reduced to square 
chains and decimal parts, and the lines to be in chains and deci- 
mals. 

A. By a Line parallel to a Side. 

406. To part off a Rectangle. If the sides of the field adja- 
cent to the given side make right angles with it, the figure parted 
off by a parallel to the given side will be a rectangle, and its 
breadth will equal the required content divided by that side, as 
in Art. 398. 

If the field be bounded by a curved or zigzag line outside of the 
given side, find the content between these irregular lines and the 
given straight side, by the method of offsets, subtract it from the 
content required to be parted off, and proceed with the remainder 
as above. The same directions apply to the subsequent problems. 

407. To part off a Parallelo- 
gram. If the sides adjacent to the 
given side be parallel, the figure 
parted off will be a parallelogram, 
and its perpendicular width, CE, 

* The given lines will be represented by fine full lines, the lines of construction 
by broken lines, and the lines of the result by heavy full lines. 

18 





Fig. 


285. 




c 








/ 1 


N \ 




"/ 


/ \ 




\ 


y 








B 



268 



LAND-SUR VEYINQ. 



will be obtained as above. The length of one of the parallel sides, 



as AC =- r = 



ABDC 



sin. A AB . sin. A 



Fig. 286. 



408. To part off a Trapezoid. When the sides of a field ad- 
jacent to the given side are not parallel, the figure parted off will 
be a trapezoid. 

When the field or figure is given on the ground, or on a plat, 

begin as if the sides were parallel, 
dividing the given content by the 
base AB. The quotient will be 
an approximate breadth, CE, or 
D F ; too small if the sides con- 
verge, as in the figure, and vice 
versa. Measure CD. Calculate 
the content of ABDC. Divide 
the difference of it and the required content by C D. Set off the 
quotient perpendicular to C D (in this figure, outside of it), and it 
will give a new line, G H, a still nearer approximation to that de- 
sired. The operation may be repeated, if found necessary. 




409. When the field is given by bearings, deduce from them 
the angles at A and B. The required sides will 
then be given by these formulas : 

/(A B , _ 2xABCD.sin. ( A+B)\ 

V \ sir* A oin Ti / * 



Fig. 287. 



CD 



AD = (AB-CD) 



sin. A . sin. B 
sin. B 




sin. (A + B) * 
sin. A 



v ' sin. (A + B) 

Demonstration. Produce B C and A D to meet in E. 
By similar triangles, 

ABE :DCE :: AB 2 : D C 2 . 
A B E - D C E : A B E :: A B 2 - D C Q : A B 2 
Now ABE — DCE = ABCD; also, by Art. 61, note, 

ABE = AB 2 . f- A ;f\ B . 
2. sin. (A + B) 

The above proportion, therefore, becomes 



PARTING OFF LAND. 



269 



ABCD : AB 2 . 



sin. A . sin. B 



2 . sin. (A + B) 
Multiplying extremes and means, cancel- 
ing, transposing, and extracting the 



:: AB- 



CD 2 : A B 2 . 
Fig. 288. 



square root, we get C 



2. AB D . sin. (A + B) 



]■ 



When A + B > 180°, sin. (A + B) is 
negative, and therefore the fraction in 
which it occurs becomes positive. 

C F being drawn parallel to D A, we have 
sin. B 



:--E 



AD=FC= FB . 



= (A B - D) 



sin. B C F 
sin. B 



= FB. 



sin. B 



sin. (180° - A 
BO = (AB-OD) 



-B) 
sin. A 



sin. (A + B) s ' sin. (A + B) 

When the sides A D and B C diverge, instead of converging, as 
in the figure, the negative term, in the expression for C D, becomes 
positive ; and, in the expressions for both A D and B 0, the first 
factor becomes (C D — A B). 

The perpendicular breadth of the trapezoid =AD. sin. A ; 
or = B . sin. B. 

Example. Let A B run north, six chains ; AD, N. 80° E. ; 
B C, S. 60° E. Let it be required to part off one acre by a fence 
parallel to A B. Here AB = 6 -00, ABCD = 10 square chains, 
A = 80°, B = 60°. Ans. C D = 4-57, A D = 1-92, B C = 2*18, 
and the breadth = 1 '89. 

The figure is on a scale of 4 chains to an inch = 1 : 3168. 

B. By a Line perpendicular to a Side. 
410. To part off a Triangle. Let F Gr be the required line. 

When the field is given on the 
ground, or on a plat, at any point, 
as D, of the given side AB, set out 
a " guess-line, " D E, perpendicular 
to A B, and calculate the content of 
D E B. Then the required distance 
BF, from the angular point to the foot of the desired perpen- 
dicular = b d i//ii^y 









Fig. 289. 




270 LAND-SURVEYING. 

Since similar triangles are as the squares of their homologous sides, 
BDE: BFG:: B D 2 : BF 2 ; whence BF = BD j/(— — -). 

Example. Let B D = 30 chains ; E D = 
12 chains ; and the desired area = 24 -8 acres. 
Then B F = 35 '22 chains. 

The scale of the figure is 30 chains to 1 
inch = 1 : 23760. 

When the field is given by bearings, find 
the angle B from the bearings ; then is 




"VC-SfO- 



. Example. Let B A bear S. 75° E., and BCK 60° E., and let 
five acres be required to be parted of from the field by a perpen- 
dicular to B A. Here the angle B = 45°, and B F = 10*00 
chains. 

The scale of the figure is 20 chains to 1 inch = 1 : 15840. 

411. To part off a Quadrilateral. Produce the converging 
sides to meet at B. Calculate the 

content of the triangle HKB, 
whether on the ground or plat, or 
from bearings. Add it to the con- 
tent of the quadrilateral required B ^;::1....- .... 
to be parted off, and it will give 

that of the triangle F G B, and the method of the preceding case 
can then be applied. 

412. To part off any Figure. If the field be very irregularly 
shaped, find by trial any line which will part off a little less than 
the required area. This trial-line will represent H K in the pre- 
ceding figure, and the problem is reduced to parting off, accord- 
ing to the required condition, a quadrilateral, comprised between 
the trial-line, two sides of the field, and the required line, and con- 
taining the difference between the required content and that parted 
off by the trial-line. 



PARTING OFF LAND. 



271 



0. By a Line running in any Given Direction. 

413. To part off a Triangle. By construction, on the ground 
or the plat, proceed nearly as in Art. 410, setting out a line in the 
required direction, calculating the triangle thus formed, and ob- 
taining B F by the same formula as in that article. 



C B A and G F B ; then is B F 



^ 




414. If the field be given by bearings, find from them the angles 

2X B F G sin. (B + F) \ 

sin. B . sin. F / ' 

Example. Let B A bear S. 30° E. ; B C, 
N. 80° E. ; and a fence be required to run from 
some point in B A, a due north course, and to 
part off one acre. Required the distance from 
B to the point F, whence it must start. Ans. 
The angle B = 70°, and F = 30°. Then B F 
= 6 -47. 

The scale of Fig. 292 is 6 chains to 1 inch 
= 1 : 4752. 

415. To part off a Quadrilateral. Let it be 

required to part off, by a line running in a given direction, a 
quadrilateral from a field in which are given the side A B, and 

the directions of the two 
other sides running from A 
and from B. 

On the ground or plat 
produce the two converging 
sides to meet at some point 
E. Calculate the content of 
the triangle ABE. Meas- 
ure the side A E. From 
ABE subtract the area to 
be cut off, and the remain- 
der will be the content of 
the triangle ODE. From A set out a line A F parallel to the 
given direction. Find the content of A B F. Take it from 



Fig. 293. 



„-,E 




272 LAKD-SURVETIKG. 

ABE, and thus obtain A E E. Then this formula, E D = A E 

^FXE' wil1 fix the point J) ' since AD = AE ~ ED - 

When the field and the dividing line are given by bearings, 
produce the sides as in the last article. Eind all the angles from 
the bearings. Calculate the content of the triangle ABE, by the 
formula for one side and its including angles. Take the desired 
content from this to obtain CDE. Calculate the side A E = A B 

'2XCDE. sm. D C E\ 



Then is AD = AE 



-A 



sin. E . sin. CDE 



Demonstration. Since triangles which have an angle in each equal, are 
as the products of the sides about the equal angles, we have 
ABE: ODE:: AE xBE:OExDE. 

ABE = |-.AB^ slD - A ; sin R B . AE=AB.^l|. 
* sin. (A + B) sm. E 

BE = AB.^. 0E = DE.5B^£|. 

sin. E sm. DCE 

Substituting these values in the preceding proportion, canceling the common 
factors, observing that sin. (A + B) = sin. E, multiplying extremes and 

i-;r m- t-nv // 2 • C D E . sin. D CE \ 

means, and dividing, we get DE=i/ — : — : — _ ■ I • 

r V sm. E . sm. CDE/ 

Example. Let D A bear S. 20J° W. ; A B, 2s T . 51|° W., 8-19 ; 
B C, N. 73 1° E ; and let it be required to part off two acres by a 
fence, D C, running K 45° W. Am. A B E = 32-56 square 
chains ; whence C D E = 12-56 square chains. Also, A E = 8 -37 ; 
and, finally, AD = 8*37 — 5-51 = 2*86 chains. 

The scale of Eig. 293 is 5 chains to 1 inch = 1 : 3960. 

If the sum of the angles at A and B were more than two right 
angles, the point E would lie on the other side of A B. The neces- 
sary modifications are apparent. 

416. To part off any Figure. Proceed in a similar manner to 
that described in Art. 412, by getting a suitable trial-line, pro- 
ducing the sides it intersects, and then applying the method just 
given. 

D. By a Line staetino feom a Given Point in a Side. 

417. To part off a Triangle. Let it be required to cut of from 
a corner of a field a triangular space of given content, by a line 
starting from a given point on one of the sides, A in the figure, 



PARTING OFF LAND. 



273 



Fig. 294. 



~-tC 




the base, A B, of the desired triangle being thus given. If the 
field be given on the ground or on 
a plat, divide the given content 
by half the base, and the quotient 
will be the height of the triangle. 
Set off this distance from any 
point of A B, perpendicular to it, 
as from A to C ; from set out 
a parallel to A B, and its inter- 
section with the second side, as at 
D, will be the vertex of the required triangle. 

Otherwise : Divide the required content by half of the perpen- 
dicular distance from A to B D, and the quotient will be B D. 

If the field be given by the bearings of two sides and the 
length of one of them, deduce the angle B (Fig. 294) from the 

, . mu • dt^ 2XABD 

bearings. Then isBD= T -=, = — =, . 

to A B . em. B 

If it is more convenient to fix the point D, by the second 
method, that of rectangular co-ordinates, we shall have B E = 
B D . cos. B ; and E D = B D . sin. B. 

The bearing of A D is obtained from the angle BAD, which is 
ED ED 



known, since 



= tan. BAD. 



EA" AB-BE 

Example. Eighty acres are to be set off from a corner of a 
field, the course AB being N. 80° W., sixty chains ; and the bear- 
ing of B D being 1ST. 70° E. Ans. B D = 53'33 ; B E = 46*19 ; 
E D = 26-67 ; and the bearing of A D, N". 17° 23' W. 

The scale of Fig. 294 is 40 chains to 1 inch = 1 : 31680. 

2 A B D 

If the field were right-angled at B, of course D B = • — . ~ . 

418. To part off a Quadrilateral. Imagine the two converging- 
sides of the field produced to meet, as in Art. 415. Calculate the 
content of the triangle thus formed, and the question will then be 
reduced to the one explained in the last two articles. 

419. To part off any Figure. Proceed as directed in Art. 416. 
Otherwise, proceed as follows : 



274 LAND-SURVEYING. 

The field being given on the ground or on a plat, find on which 

side of it the required line will 

Fig 295. 

end, by drawing or running 

" guess-lines " from the given 

point to various angles, and 

\ roughly measuring the content 

-"•^v thus parted off. If, as in the 

_^>e figure, A being the given point, 

/ the guess-line AD parts off less 
than the required content, and 
A E parts off more, then the desired division-line A Z will end in 
the side D E. Subtract the area parted off by A D from the re- 
quired content, and the difference will be the content of the tri- 
angle ADZ. Divide this by half the perpendicular let fall from 
the given point A to the side D E, and the quotient will be the 
base, or distance from D to Z. 

Or, find the content of A D E and make this proportion : 
ADE : ADZ ::DE : D Z. 

The field being given by bearings and distances, find as be- 
fore, by approximate trials on the plat, or otherwise, which 
side the desired line of division will terminate in, as D E in 
the last figure. Draw AD. Find the latitude and departure 
of this line, and thence its length and bearing. Then calculate 
the area of the space this line parts off, A B C D in the figure, 
by the usual method, explained in Part I, Chapter III. Sub- 
tract this area from that required to be cut off, and the remainder 
will be the area of the triangle ADZ. Then, as in Art. 415, D Z = 

2ADZ 
AD. sin. ADZ" 

This problem may be executed without any other table than 
that of latitudes and departures, thus : Find the latitude and de- 
parture of D A, as before, the area of the space A B C D, and 
thence the content of A D Z. Then find the latitude and departure 
of E A, and the content of A D E. Lastly, make this proportion : 
ADE : ADZ ::DE : D Z.* 

* The problem may also be performed by making the side on which the division- 



PARTING OFF LAND. 



275 



Example. In the field ABODE, etc., part of which is shown 
in Fig. 295 (on a scale of 4 chains to 1 inch = 1 : 3168), one acre 
is to be parted off on the west side, by a line starting from the 
angle A. Eequired the distance from D to Z, the other end of this 
dividing line.* 

The only courses needed are these : AB, N. 53° W., 1*55 ; B C, 
N. 20° E, 2-00; CD, N. 53J° E., 1-32 ; DE, S. 57° E., 5-79. A 
rough measurement will at once show that A B C D is less than an 
acre, and that ABODE is more ; hence the desired line will fall 
on D E. The latitudes and departures of A B, B 0, and C D are 
then found. From them the course A D is found to be N. 8° V 22" 
E., 3*634. The content of ABOD will be 3'19 square chains. 
Subtracting this from one acre, the remainder, 6*81 square chains, 
is the content of A D Z. A P = 3 -63 X sin. 65° = 3 -29. Dividing 
A D Z by half of this, we obtain D Z = 4*14 chains. 

By the second method, the latitude and departure of D A, the 
area of A B C D, and of A D Z, being found as before, we next find 
the latitude and departure of E A from those of A D and D E, and 
thence the area of AD E = 9*53. Lastly, we have the proportion 
9-53 : 6-81 : : 5*79 : D Z = 4*14, as before. 



E. By a Line passing through a Given Point within the Field. 



420. To part off a Triangle. 
Let P be a point within a field 
through which it is required to 
run a line so as to part off from 
the field a given area in the form 
of a triangle. 

When the field is given on 

line is to fall, a meridian, and changing 
the bearings. The difference of the new 
departures will be the departure of the 
division-line. Its position can then be 
easily determined. 

* If the whole field has been surveyed 
and balanced, the balanced latitudes and 
departures should be used. We will here 
suppose the survey to have proved per- 
fectly correct. 



Fig. 296. 



M' B 




276 LAND-SURVEYING, 

the ground or on a plat, the division can be made by construction, 
thus : Divide the given area by half of the perpendicular distance 
from P to A C, and set off the quotient from C to G-. Bisect 
G C in H. From P draw P E, parallel to the side B C. On H E 
describe a semicircle. On it set off E K = E C. Join K H. Set 
off H L = H K. The line L M, drawn from L through P, will 
be the division-line required.* If HK be set off in the contrary 
direction, it will fix another line 1/ P M', meeting C B produced, 
and thus parting off another triangle of the required content. 

Demonstration. By construction, G P C = the required content. Xow, 

GPO = (xDO, since they have the same base and equal altitudes. We have 

now to prove that LMO= GDC. These two triangles have a common 

angle at 0. Hence, they are to each other as the rectangles of the adjacent 

sides i e 

GDC:LMC::GC x CD::LCx CM. 

Here C M is unknown, and must he eliminated. TVe obtain an expression for 

it by means of the similar triangles LMO and L E P, which give 

LE:LC::EP = CD:CM. 

CD x LC 

Hence, CM= — — . Substituting this value of AI in the first pro- 

L E 

portion, and canceling C D in the last two terms, we get 

GDC:LMC::GC: ^"; orGDC:LMC::GCxLE:LC 2 . 
L E 

LC 2 = (LH + HC) 2 = LH 2 + 2LH x H C + H C 2 . 

But, by construction, 

LH 2 =HK 2 =HE 2 -EK 2 =HE 2 -EC 2 = (HE + EC)(HE-EC) = HC(HE-EC). 

Also, G C = 2 H C ; and L E = L H + H E. 

Substituting these values in the last proportion, it becomes 



GDC: LMC:: 2. HC(LH + HE) 
: : 2 L H + 2 H E 



HC(HE-EO+2LHxHC + HC 2 . 

HE-EC + 2LH + HC. 

HE-EC + 2LH + HE + EC. 

2 HE + 2LH. 

The last two terms of this proportion are thus proved to be equal. There- 
fore, the first two terms are also equal — i. e., LMC = GDC = the required 
content. 

Since HK= V (H E 2 — E K 2 ), it will have a negative as well as a posi- 
tive value. It may therefore be set off in the contrary direction from L — 
i. e., to I/. The line drawn from L/ through P, and meeting C B produced 
beyond B, will part off another triangle of the required content. 

Example. Let it be required to part off 31-175 acres by a fence 
passing through a point P, the distance P D of P from the side 

* As some lines in the figure are not used in the construction, though needed for 
the demonstration, the student should draw it himself to a large scale. 



PARTING OFF LAND. 277 

B C, measured parallel to A 0, being 6 chains, and DC 18 chains. 
The angle at C is fixed by a " tie-line " A B = 48-00, B C being 
42-00, and A being 30-00. Ans. 
CL= 27-31 chains, or CL' = 7*69 
chains. 

The figure is on a scale of 20 
chains to 1 inch = 1 : 15840. 

If the angle of the field and the 
position of the point P are given by 
bearings or angles, proceed thus : 
Find the perpendicular distances, P Q 
and P R, from the given point to 
the sides, by the formulas P Q = P C . sin. POQ; and PE = 
P . sin. P C E. Let PQ = ^PR=j?, and the required con- 




o V \f p sin. LCM/' 



tent = c. Then C L 

P 

Demonstration. Suppose the line L M drawn. Then, by Art. 61, note, 

the required content, c = }.OLxCM. sin. LCM. This content will also 

equal the sum of the two triangles L C P and M P — i. e.,« = }.CLxp + 

2 c 

I.CMxff. The first of these equations gives O M = — — : — . 

2 * O L . sin. L M 

Substituting this in the second equation, we have 

c = ^.CLxp + — - Cq y . 

4 O L . sin. LOM 

Whence, \p . C L 2 . sin. LCM+q=c.OL. sin. LCM. 

Transposing and dividing by the coefficient of C L 2 , we get 

CL<- 2 - C .CL = - 



p p . sin. C L M 

CL= C |./Y- 2cy \ 

P V \p* p . sin. L C Mj ' 
If the given point is outside of the lines C L and C M, conceive the de- 
sired line to be drawn from it, and another line to join the given point to 
the corner of the field. Then, as above, get expressions for the two triangles 
thus formed, and put their sum equal to the expression for the triangle which 
comprehends them both, and thence deduce the desired distance, nearly as 
above. 

Example. Let the angle LCM = 82°. Let it be required to 
part off the same area as in the preceding example. Let PC = 
19-75, PCQ=17°30f, P C R = 64° 29J-'. Required C L. Ans. 
,PQ= 5-94, PR = 17-82, and therefore, by the formula, OL = 



278 



LAND-SUE VEYING. 



Fig. 298. 




27 '31, or CL' = 7*69; corresponding to the graphical solution. 
The figure is on the same scale. 

If the given point were without the field, the division-line could 
be determined in a similar manner. 

421. To part off a Quadrilateral. Conceive the two sides of the 
field which the division-line will intersect, D A and C B, produced 
till they meet at a point G, not shown in 

the figure. Calculate the triangle thus 
formed outside of the field. Its area, in- 
creased by the required area, will be that 
of the triangle E F G. Then the problem 
is identical with that in the last article. 
The following example is that given in 
Gummere's "Surveying." The figure rep- 
resents it on a scale of 20 chains to 1 inch 
= 1 : 15840. 

Example. A field is bounded thus : N. 14° W., 15*20 ; N. 70£° 
E., 20-43 ; S. 6°E., 22*79 ; K 86J° W., 18-00. A spring within 
it bears from the second corner S. 75° E., 7*90. It is required to 
cut off 10 acres from the west side of the field by a straight fence 
through the spring. How far will it be from the first corner to 
the point at which the division-fence meets the fourth side ? Ans. 
4-6357 chains. 

422. To part off any Figure. Let it be required to part off from 

a field a certain area bv a line 

Fig 299 

passing through a given point 

P within the field. Eun a 
guess - line A B through P. 
Calculate the area which it 
parts off. Call the difference 
between it and the required 
area = d. Let C D be the de- 
sired line of division, and let 
P represent the angle, APC 
or B P D, which it makes with the given line. Obtain the angles 
P A C = A, and P B D = B, either by measurement, or by de- # 



B J) 




PARTING OFF LAND. 279 

duction from bearings. Measure P A and P B. Then the desired 

angle P will be given by the following formula : 

/ A P 2 — . V> V'\ 

Cot. P = - i (cot. A + cot. B - ^~Yd^) ± 

r /A P 3 . cot. B-BF. cot. A . . _ , 

W 2d cot. A. cot. B + 

t (cot.A+cot.B- Apa 2 - BF )T 

If the guess-line be run so as to be perpendicular to one of the 
sides of the field, at A, for example, the preceding expression re- 
duces to the following simpler form : 

Cot. P = - i (cot. B - AF '~/ F ) ± 

/ r A P' . cot. B , , ( . _ AF-BP' V] 
V V 2d + H oot - B 2d ) -I • 

Demonstration. The difference d, between the areas parted off by the 
guess-line A B, and the required line D, is equal to the difference between 
the triangles A P C and B P D. 

By Art. 61, note, the triangle AP C = * . A P 2 . ym - A • sm - p ; 

sin. (A + P) 

Similarly, the triangle B P D = £ . B P 2 sin - B • sm - p g 

sin. (B + P) 

. rJ — i A P2 sin. A sin. P R sin. B . sin. P 
••^- i,AP sin.(A + P)" iBP ' sin. (B + P) ' 
By the expression for sin. (a + &) [Trigonometry, Art. 8], we have 

d=lAP\-. 8in - A - 8i "- P JBP" sin. B. sin. P 



sin. A . cos. P + sin. P . cos. A sin. B . cos. P + sin. P. cos. B 

Dividing each fraction by its numerator, and remembering that —^ — = 

sin. a 
= cot. «, we have 

£AP* _ |BF 

cot. P + cot. A cot. P + cot. B ' 
For convenience, let p = cot. P ; a = cot. A ; and i = cot. B. The above 
equation will then read, multiplying both sides by 2, 

p + A p + b 

Clearing of fractions, we have 

2dp*+2dap + 2dbp + 2 dab = p . AF*+ 1) . AP 2 — p . B P 2 -a . B P 2 . 

Transposing, dividing through by 2 d, and separating into factors, we get 

. / . AP 2 -BP 2 \ 5.AF-«.BP a 
P + (« + * ^— > = U ah. 

AP-a. BP 2 



. / , , ap 2 -bp 2 \ L . ri. 



2d 



280 



LAND-SUR VEYWG. 



— ab + -1- ( a + b - 



AP-BP ! 
2d 



)']• 



If A = 90°, cot. A = 0; and the expression reduces to the simpler form 
given in the article. 

Example. It was required to cut off from a field twelve acres 
by a line passing through a spring P. A guess-line, A B, was run 
making an angle with one side of the field, at A, of 55°, and with 
the opposite side, at B, of 81°. The area thus cut off was found 
to be 13*10 acres. From the spring to A was 9*30 chains, and to 
B 3-30 chains. Required the angle which the required line, C D, 
must make with the guess-line, A B, at P. Ans. 20° 45' ; or 
- 86° 25'. The heavy broken line, C D', shows the latter. 

The scale of the figure is 10 chains to 1 inch = 1 : 7920. 

If the given point were outside of the field, the calculations 
would be similar. 

F. By the Shoetest Possible Line. 
423. To part off a Triangle. Let it be required to part off a 
triangular space, BDE, of given content, from 
the corner of a field, ABO, by the shortest pos- 
sible line, D E. 

From B set off B D and B E each equal to 
7/ 2 B DE \ The Hne D E tlmg obtained ^pi 
r \ sin. B J 

be perpendicular to the line, B F, which bi- 
sects the angle B. The length of D E = 
V (2 . D B E . sin. B) 
cos. \ B 

Demonstration Conceive a perpendicular, B F, to be let fall from B to 

the required line D E. Let B represent the angle DBE, and /3 the unknown 

angle DBF. The angle B D F = 90° — ; and the angle B E F = 90° — 

(B — |3) = 90° — B + £. By Art. 61, note, the area of the triangle DBE 

_ DEa sin. BDE. sin. BED _ sin. (90°- /3) sin. (90°-B+/3) 

~ * " sin. (BDE+BED)"'" 

^ „. 2xDBEx sin. B 

Hence, DE- 



Fig. 300. 




sin. B 
2 x D B E x sin. B 



sin. (90°— /3) . (sin. (90°— B + /8) cos. £ . cos. (B — /3) 

Now, in order that D E may be the least possible, the denominator of 

the last fraction must be the greatest possible. It may be transformed, by 

the formula, cos. a . cos. o = $ cos. (a + o) + i . cos. (a — o) [Trigonometry, 

Art. 8], into £ cos. B -f i . cos. (B — 2 £). Since B is constant, the value of 



PARTING OFF LAND. . 281 

this expression depends on its second terra, and that will be the greatest pos- 
sible when B — 2 /3 = 0, in which case /3 = \ B. 

It hence appears that the required line D E is perpendicular to the line, 
B F, which bisects the given angle B. This gives the direction in which 
D E is to be run. 

Its starting-point, D or E, is found thus : The area of the triangle D B E 
= \ B D . B E . sin. B. Since the triangle is isosceles, this becomes 

DBE =iBD 2 .sin.B; whence BD = j/7 2 D B - \ . 

D E is obtained from the expression for D E 2 , which becomes, making /3= J- B, 

^_, a 2xDBExsin. B . n „ V (2 . DBE . sin. B) 

D E 2 = — , wlience, DE= — . 

cos. % B . cos. i B cos. i B 

Example. Let it be required to part off 1 *3 acre from the cor- 
ner of a field, the angle, B, being 30°. Ans. B D = B E = 7 '21 ; 
and DE = 3-73. 

The scale of the figure is 10 chains to 1 inch = 1 : 7920. 

G. Land of Variable Value. 

424. Let the figure represent a field in which the land is of two 
qualities and values, divided by the " quality-line " 
E F. It is required to part off from it a quantity IG * " 

of land worth a certain sum, by a straight fence 
parallel to A B. 

Multiply the value per acre of each part by 
its length (in chains) on the line A B, add the 
products, multiply the value to be set off by 10, 
divide by the above sum, and the quotient will be 
the desired breadth, B C or A D, in chains. 

Demonstration. Let a = value per acre of one portion of the land, 
and o that of the other portion. Let x == the width required, B C or A D. 

Then the value ofBCFE=ax x * BE , and the value ofADFE = 5x 
x x A E 



10 

Putting the sum of these equal to the value required to be parted off, we 

, , . value required x 10 
obtain x = — . 

Example. Let the land on one side of E F be worth 1200 per 
acre, and on the other side $100. Let the length of the former, 
B E, be 10 chains, and E A be 30 chains. It is required to part off 



282 



LAND-SUE VEYING. 



a quantity of land worth 87,500. Ans. The width of the desired 
strip will be 15 chains. 

The scale of the figure is 40 chains to 1 inch = 1 : 31680. 

If the " quality-line " be not perpendicular to A B, it may be 
made so by "giving and taking," or as in the article following 
this one. 

The same method may be applied to land of any number of 
different qualities ; and a combination of this method with the 
preceding problems will solve any case which may occur. 

H. Straightening Cbooked Fences. 

425. It is often required to substitute a straight fence for a 
crooked one, so that the former shall part off precisely the same 
quantity of land as did the latter. This can be done on a plat by 

Fig. 302. 




the method given in Art. 70, by which the irregular figure 1...2... 
3... 4... 5 is reduced to the equivalent triangle 1...5...3', and the 
straight line 5... 3' therefore parts off the same quantity of land on 
either side as did the crooked one. The distance from 1 to 3', as 
found on the plat, can then be set out on the ground and the 
straight fence be then ranged from 3' to 5. 

The work may be done on the ground more accurately by run- 

Fig. 303. 




ning a guess-line, A C, Fig. 303, across the bends of the fence 
which crooks from A to B, measuring offsets to the bends on each 



DIVIDING UP LAND. 283 

side of the guess-line, and calculating their content. If the sums 
of these areas on each side of A chanced to be equal, that would 
be the line desired ; but if, as in the figure, it passes too far on one 
side, divide the difference of the areas by half of A 0, and set off 
the quotient at right angles to A C, from A to D. DC will then 
be a line parting off the same quantity of land as did the crooked 
fence. If the fence at A was not perpendicular to A 0, but oblique, 
as A E, then from D run a parallel to A C, meeting the fence at E, 
and E C will be the required line. 

DIVIDING UP LAND. 

426. Most of the problems for " dividing up " land may be 
brought under the cases in the preceding articles, by regarding one 
of the portions into which the figure is to be divided as an area 
to be "parted off" from it. Many of them, however, can be most 
neatly executed by considering them as independent problems, and 
this will be here done.. They will be arranged, first, according 
to the simplicity of the figure to be divided up, and then sub- 
arranged, according to the manner of the division. 

Division of Triangles. 

427. By Lines parallel to a Side. Suppose that the triangle 
ABO is to be divided into two equivalent 

parts by a line parallel to AC. The desired ' ' E 



point, D, from which this line is to start, 

will be obtained by measuring BD=AB^ t> / \ E 

\. So, too, E is fixed by B E = B V h ^ -X 

Generally, to divide the triangle into two 
parts, B D E and A C E D, which shall have to each other a ratio 

m 



= m : n, we have BD = ABj/ 



G-- 



7n-\- n 

Fig. 305. This may be constructed thus : Describe a 

semicircle on A B as a diameter. From B set off 



B P = — ; — . B A. At P erect a perpendicular 

/ \ meeting the semicircle at G. Set off B G from 

c B to D. D is the starting-point of the division- 




284 LAND-SURVEYING. 

line required. In the figure, the two parts are as 2 to 3, and 
B F is therefore = f B A. 

To divide the triangle ABO into five equivalent parts, we 

should have, similarly, BD=AB^{; 

BD'=ABV1;BD' = ABV}; BD'" 

= ABV* 

The same method will divide the tri- 
angle into any desired number of parts 
having any ratios to each other. 

428. By Lines perpendicular to a Side. 

Suppose that A B C is to be divided into 
two parts having a ratio = m : n, by a line perpendicular to A C. 
Let E F be the dividing line whose po- 
sition is required. Let B D be a per- 
pendicular let fall from B to A C. 

Then is A E = y (a C X A D X 
^—)- Inthisfigure,AFE:EFBC 

: : m : n : : 1 : 2. 

If the triangle had to be divided into two equivalent parts, the 
above expression would become AE= ^(JACxAD). 

Demonstration. By hypothesis, A E F : E F B C : : m : n ; whence A E F : 

k -n r> jA-nr^ a-t.^. m AC X DB W 

ABC: : m: m + n; and AEF=ABC = . . 

in + n 2 m + ?i 

Also, AEF = |.AE xEF. 

The similar triangles A E F and A BD giye AD:DB::AE;EF = 

D B x A E 

— . The second expression for A E F then becomes AEF = 

|AE. DB X t^ E • Equating this with the other yalue of A E F. we hare 
A L) 

ACxDB m AE 2 xDB , AT . / / . ^ lTA m 



Fig 


307. 


B 


r 


i\ 



m + n 2 . AD 



; whence A E = a/(A CxADx -^— V 



429. By Lines running in any Given Direction. Let a triangle, 
A B C, be given to be divided into two parts, having a ratio = m : n, 
by a line making a given angle with a side. Part off, as in Art. 

413 or 414, Fig. 292, an area B F G = -^- .ABC. 



DIVIDING UP LAND. 



285 



Fig. 308. 



430. By Lines starting from an Angle. Divide the side oppo- 
site to the given angle into the required num- 
ber of parts, and draw lines from the angle 
to the points of division. In the figure the 
triangle is represented as being thus divided 
into two equivalent parts. 

If the triangle were required to be divided 
into two parts, having to each other a ratio = 

m 




m : n, we should have A D = A Cl- 



aud D C = A C 



Fig. 309. 




m-\- n' m + n' 

If the triangle had to be divided into three 
parts which should be to each other : : m : n 

m 



: p, we should have A D = A C 

D E = A C 
P 



m 



m -\-n-\-p ' 



m-\- n +p ' 

and E C = A C 



m + n -\-p' 
Suppose that a triangular field, ABC, had to be divided among 
five men, two of them to have a quarter each, and three of them 
each a sixth. Divide A C into two equal parts, one of these again 
into two equal parts, and the other one into three equal parts. 
Run the lines from the four points thus obtained to the angle B. 



431. By Lines starting from a Point in a Side. Suppose that 
the triangle A B C is to be divided into two 
equivalent parts by a line starting from a point 
D in the side AC. Take a point E in the 
middle of A C. Join B D, and from E draw 
a parallel to it, meeting A B in F. D F will 
be the dividing line required. 

The point F will be most easily obtained 
on the ground by the proportion AD :AB ::AE = iAC :AF. 

The altitude of A F D of course equals |ABC-r|AD. 

If the triangle is to be divided into two parts having any other 
ratio to each other, divide A C in that ratio, and then proceed as 




before. Let this ratio = m : n, then A F = 



ABx AC 
AD 



m 



m-\-n 



2S6 



LAND -SUB VEYING. 



Demonstration. In Fig. 310, conceive the line E B to be drawn. The 
triangle AEB = |ABC, having the same altitude and half the base ; and 
AFD = AEB, because of the equivalency of the triangles E F D and E F B, 
which, with A E F, make upAFD and A E B. 

The point F is fixed by the similar triangles A D B and A E F. 
The expression for A F, in the last paragraph, is given by the proportion, 
ABC:ADF::ABxAO:ADxAF; 
ABxACADF ABxAO m 



whence, 



AF 



AD 



ABU 



AD 



m + n 



Fig. 311. 




Next suppose that the triangle A B C is to be divided into three 

equivalent parts, meeting at D. The 
altitudes, E F and G H, of the parts 
A D E and D C G, will be obtained by 
dividing J- ABC, by half of the re- 
spective bases A D and D C. 

If one of these quotients gives an 
altitude greater than that of the tri- 
angle ABC, it will show that the two lines DE and DG would 
both cut the same side, as in Fig. 312, in 
which E F is obtained as above, and G H 
= f ABC-r-iAD. 

In practice it is more convenient to 
determine the points F and G, by these 
proportions : 

BK:AK::EF:AF;andBK:AK 
::GH: AH. 
The division of a triangle into a greater number of parts, hav- 
ing any ratios, may be effected in a similar manner. 

This problem admits of a more elegant solution, analogous to 

that given for the division into two 
parts, graphically. Divide AC into 
three equal parts at L and M. Join 
B D, and from L and M draw parallels 
to it, meeting A B and B C in E and 
G. Draw E D and G D, which will 
be the desired lines of division. The figure is the same triangle 
as Fig. 311. 

The points E and G can be obtained on the ground by measur- 




— x c 



Fig. 313. 




DIVIDING UP LAND. 



287 



Fig. 314. 




Fig. 



ing A D and A B, and making the proportion AD : A B : : J A C : 
AE. The point G- is similarly obtained. 

The same method will divide a triangle into a greater number 
of parts. 

To divide a triangle into four equivalent triangles by lines 
terminating in the sides, is very easy. 
From D, the middle point of A B, draw 
D E parallel to A C, and from F, the 
middle of A 0, draw F D and F E. 
The problem is now solved. 

432. By Lines passing through a Point 
within the Triangle. Let D be a given point (snch as a well, etc.) 

within a triangular field ABC, from 
which fences are to run so as to divide 
the triangle into two equivalent parts. 
Join A D. Take E in the middle of B 0, 
and from it draw a parallel to D A, meet- 
ing A C in F. E D F is the fence required. 
If it be required to divide a trian- 
gle into two equivalent parts by a straight line passing through 
a point within it, proceed thus : Let P be the given point. From 
P draw PD parallel to AC, and 
PE parallel to BC. Bisect AC 
at F. Join F D. From B draw 
B G parallel to D F. Then bi- 
sect G C in H. On H E de- 
scribe a semicircle. On it set off 
EK = E0. Join KH. Set off 
H L = H K. The line L M drawn 
from L, through P, will be the 
division-line required. 

This figure is the same as that 
of Art. 416. The triangle ABO 
contains 62*35 acres, and the dis- 
tance CL = 27-31 chains, as in 
the example in that article. 




Fig. 316. 




288 



LAND-SUB VEY1XG. 



433. Next suppose that the triangle A B C is to be divided into 
three equivalent parts by lines starting 
from a point D, within the triangle, 
given by the rectangular co-ordinates 
A E and E D. Let E D be one of the 
lines of division, and F and G- the 
other points required. The point F 
will be determined if A H is known ; 

A H and H F being its rectangular co-ordinates. From B let fall 

the perpendicular B K on A C. 

AK(fABC-AExED) 




Then is A H = 



The position of the 



AEXBK-ED X AK 
other point, G, is determined in a similar manner. 

Demonstration. Let AE = x,ED = |/, AH = «', HF = y', A K = ,7, 
KB = o. 

The quadrilateral A F D E, equivalent to |ABO, but which we will 
represent generally by m 2 , is made up of the triangle AFH and the trape- 
zoid FHED. 

AFH = |. x'y'.' F H E D = i (x - x 1 ) (y + y'). 

.-. A~F DE = m* = i . x' y' + % (x - x') (y + y') = $x(y + y)-$x'y. 
The similar triangles, A HF and A K B, give 

a : o : : x > : y = — . 
a 

Substituting this value of y' in the expression for m 2 , we have 

/ ox'\ 
m* = $x ly+ — J —hx'y; 

, _ a(2m*—xy) _ AK(fABC-AExED) 
wnence, x- lx _ ay - K BxAE-AKxED ' 

The formula is general, whatever may be the ratio of the area m 2 to that 
of the triangle ABC. 



Let D B, instead of D E, be 
division. Divide | ABC by 
half of the perpendicular D H, 
let fall from D to A B, and the 
quotient will be the distance 
BF. To find G, if, as in this 
figure, the triangle BDC(= 
B C X i D K) is less than £ 
ABC, divide the excess of the 



one of the required lines of 



Fig. 318. 




DIVIDING UP LAND. 289 

latter (which will be C D G) by } D E, and the quotient will 
be CG. 

Example. Let A B = 30 -00 ; B C = 45 -00 ; CA = 50 -00. Let 
the perpendiculars from I) to the sides be these: DE = 10*00; 
DH = 20-00 ;DK = 5*17J. The content of the triangle ABC 
will be 666'6 square chains. Each of the small triangles must 
therefore contain 222*2 square chains, BD being one division-line. 
We shall therefore have B F = 222*2 -f- \ D H = 22*2 chains. 
BDO=45xiX 5'17-j- = 116*4 square chains, not enough for a 
second portion, but leaving 105 *8 square chains for C D G ; whence 
C G = 21*16 chains. To prove the work, calculate the content of 
the remaining portion, GDF A. We shall find DGA = 144*2 
square chains, and AD F = 78*0 square chains, making together 
222*2 square chains, as required. 

The scale of Fig. 318 is 30 chains to 1 inch = 1 : 23760. 

434. The preceding case may be also solved graphically, thus : 
Take C L = \ A 0. Join D L, 

and from B draw B G parallel 
to D L. Join D G. It will be 
a second line of division. Then 
take a point, M, in the middle 
of B G, and from it draw a line, 
M F, parallel to D A. D F will 
be the third line of division. 
This method is neater on paper than the preceding, but less con- 
venient on the ground. 

Demonstration. In Fig. 319 D G is a second line of division, because, 
drawing B L, the triangle BLC = | ABC; and B D GC is equivalent to 
B L O, because of the common part BOLD, and the equivalency of the tri- 
angles D L G and D L B. 

To prove that D F is a third line of division, join MD and MA. Then 
B M A = | B G A. From B M A take M F A and add its equivalent MFD, 
and we have MDFB=£BGA = HABDG-BDG) = MfABC- 
BDG) = iABO- £BDG. To MDFB add MDB, and add its equiva- 
lent, \ B D G, to the other side of the equation, and we have 
MDFB + MDB = ±-ABC-£BDG + £BDG; or, BDF = |ABO. 

435. Let it be required to divide the triangle ABC into three 
equivalent triangles, by lines drawn from the three angular points 




290 



LAND-SUE YEYING. 



Fig. 320. 




Fig. 321. 




to some unknown point within the triangle. This point is now to 

be found. On any side, as A B, take 
AD = iAB. From D draw D E par- 
allel to A C. The middle, F, of D E, 
is the point required. 

If the three small triangles are not 
to be equivalent, but are to have to 
each other the ratios : : m : n : p, di- 
vide a side, AB, into parts having these ratios, and through 
each point of division, D, E, draw a 
parallel to the side nearest to it. 
The intersection of these parallels, in 
F, is the point required. In the fig- 
ure the parts ACF, ABF, B C F, 
are as 2 : 3 : 4. 

436. Let it be required to find the 
position of a point, D, situated within a given triangle, ABC, 

and equally distant from the points, 
A, B, C ; and to determine the ratios 
to each other of the three triangles 
into which the given triangle is di- 
vided. 

By construction, find the center of 
the circle passing through A, B, C. 
This will be the required point. 

ABx BCxCA 

By calculation, the distance DA=DB=DC= — r— p^ . 

J 4 X area A ±> L> 

The three small triangles will be to each other as 

the sines of their 

ADC 

BDC. 



Fig. 322. 




angles at D- 



-i. e., AD B : 
BDC:: sin. A D B : sin. ADC: sin. 
These angles are readily found, since the 
sine of half of each of them equals the opposite 
side divided by twice one of the equal distances. 

437. By the Shortest Possible Line. Let it be 
required to divide the triangle A B C by the short- A 



Fig. 323. 




DIVIDING UP LAND. 



291 



est possible line, D E, into two parts, which shall be to each 
other : : m : n ; orDBE : A B C : : m \m-\-n. 

From the smallest angle, B, of the triangle, measure along the 

sides, B A and B 0, a distance B D=B E=a/(-^- X A B X B C ). 

D E is the line required. It is perpendicular to the line B F which 
bisects the angle ABC; and it is 

= i"7( » XABXBC). 
cos. ^Bf\m-\-n / 

The formulas are obtained from Art. 419. 



Fig. 324. 
E' F' 



Division of Rectangles. 

438. By Lines parallel to a Side. Divide two opposite sides 
into the required number of parts, either equal or in any given 
ratio to each other, and the lines joining the points of division will 
be the lines desired. 

The same method is applicable to any parallelogram. 

Example. A rectangular field A B C D, measuring 15 *00 chains 
by 8*00, is bought by three men, 
who pay respectively $300, $400, 
and $500. It is to be divided 
among them in that proportion. 
Ans. The portion of the first, 
A E E'B, is obtained by making the 
proportion 300 -f 400 + 500 : 300 
: : 15-00 : AE = 3'75. E F is in 

like manner found to be 5'00 ; and FD = 6-25. BE' is made 
equal to A E ; E' F' to E F ; and F' C to F D. Fences from E to 
E', and from F to F', will divide the land as required. 

The scale of the figure is 10 chains to 1 inch = 1 : 7920. 

The other modes of dividing up rectangles will be given under 
the head of " Quadrilaterals," Art. 443, etc. 

Division of Trapezoids. 

439. By Lines parallel to the Bases. Given the bases and a 
third side of the trapezoid, A B D, to be divided into two parts, 
such that BCFE:EFDA::ra:rc. 



292 LAND-SUE V EYING. 

The length of the desired dividing line, 



EF 



*/(' 



m x AD 8 + w XBC ] 
m + n 



— i The distance BE= a T) — B f 1 — 

i Demonstration. In Fig. 325, conceive the sides 

A B and D C, produced, to meet in some point P. 
1 Then, by reason of the similar triangles, A DP: 

p B C P : : A D 2 : B C 2 , whence, by " division," 

! ADP-BCP = ABCD:BCP::AD 2 - 

I B C 2 : B 2 . 

In like manner, comparing E F P and B C P, 
1 we get E B C F : B C P : E F 2 - B C 2 : B C 2 . 

j Combining these two proportions, we have 

1 ABCD:EB€F::AD 2 -BC 2 :EF 2 -BC 2 ; 

J or, m + n; m:: A D 2 - B C 2 : E F 2 - B C 2 . 

Hi D Whence, (m + n) E F 2 - m . B C 2 - n B C 2 = m . 

AD 2 - m. BC 2 ; 
. EF= // mxAD 2 + nxBC 2 \ 
V \ m + n / 

Also, from the similar triangles formed by drawing B L parallel to C D, we 
have 

— BE=B^ = AB S ^BO), 

Example, Let A D = 30 chains ; B C = 20 chains ; and A B = 
54J chains ; and the parts to be as 1 to 2 ; required E F and B E. 
Ans. EF = 23*80; and BE = 20*65. 

The figure is on a scale of 30 chains to 1 inch = 1: 23760. 

440. Given the bases of a trapezoid, and the perpendicular dis- 
tance, B H, between them ; it is required to divide it as before, 
and to find E F, and the altitude, B G, of one of the parts. Let 

T> p w T> TT 

BOFE:EFDA::w:w. Then B G = - AJ) _ B Q + 

/r m 2X ABCDxBH /B O X B H \ a 1 
Vl m -+n X AD^FO + VAD-BC7 J" 

EF^C + BGX^ . 

Demonstration. Let B E F C = m .ABCD = a; let B C = ?> ; B H 

m + n 

= h; and A D — B O = c. Also, let B G = x : and E F = y. Draw B L 

parallel to C D. By similar triangles, AL:EK::BA:BE::BH: 



DIVIDING UP LAND. 



293 



BG; or, AD-BC:EF-BC::BH:BG;i. e., c : y-b :: h : x; 

, ■ h(y-h) 

whence x = — . 

c 

Also, the area BEFC = a = $.BG(EF + BO) = £a;(y + &); whence 

2 a 

y = 5. 

Substituting this value of y in the expression for a?, and reducing, we obtain 



z 2 + 



2 &A 



2ali , bh //2ah £ 2 A 2 \ 
= ; whence we have x= ±4/ ( 1 „— )• 

6 C V V C C 2 / 



The second proportion above gives y — b = — ; whence y — ~b + — . x. 

it fi 

Eeplacing the symbols by their lines, we get the formulas in the text. 

Example. Let A D = 30*00 ; B C = 20-00 ; B H = 54-00 ; and 
the two parts to be to each other : : 46 : 89. 

The above data give the content of A B C D = 1,350 square 
chains. Substituting these numbers in the above formula, we ob- 
tain B G = 20-96, and E F = 23-88. 

441. By Lines starting from Points in a Side. To divide a 
trapezoid into parts equivalent, or having any ratios, divide its par- 
allel sides in the same ratios, and join the corresponding points. 

If it *be also required that the division-lines shall start from 
given points on a side, proceed 
thus : Let it be required to 
divide the trapezoid A B O D 
into three equivalent parts by 
fences starting from P and Q. 
Divide the trapezoid, as above 
directed, into three equivalent 
trapezoids by the lines E E 
and G H. These three trap- 
ezoids must now be transformed, thus : Join E P, and from F 
draw F K parallel to it. Join P R, and it will be one of the divis- 
ion-lines required. 

The other division-line, Q S, is obtained similarly. 

442. Other Cases. For other cases of dividing trapezoids, apply 
those for quadrilaterals in general, given in the following articles.* 



Fig. 326. 
E S G 




* If a line be drawn joining the middle points of the parallel bases of a trape- 



294 



LAND-SUR VEYING. 



Fig. 327. 
G 



Division of Quadrilaterals. 
443. By Lines parallel to a Side. Let ABCD be a quadri- 
lateral which it is required to divide, by a line E E, parallel to 

A D, into two parts, BEFC 
and E F D A, which shall be 
to each other as m : n. Pro- 
long A B and CD to inter- 
sect in G. Let a be the 
area of the triangle A D G, 
obtained by any method, 
graphical or trigonometrical, 
and a' = the area of the tri- 
angle B C G, obtained by 
J) subtracting the area of the 
given quadrilateral from that 
ma-\-na! s 



of the triangle A D G. Then G K 



Hav- 



y \(m + n) aJ 

ing measured this length of G K from G on G H, set off at K a 
perpendicular to G K, and it will be the required line of division. 

Demonstration. In Fig. 327, since E F is parallel to A D, we have 
ADG : EGF : : GH 2 : GK 2 . EGF is made up of the triangle B C G = a', 



and the quadrilateral B E F = 



m 



. ABCD= 



m 



m + n 



{a -a 1 ). 



m + n 
Hence the above proportion becomes 

a:a' + — — (d-a')::Gff:GK s ; or, 
m + n 

(m + n)a:ma + na' ::GH 2 : GK 2 ; whence GE = GH a/ ( ™ a + \ a ' ) . 

r \{m+ri)a/ 

G E is given by the proportion GH:GK::GA:GE=GA. ^5 . 

G H 

In Fig. 328, the division into p parts is founded on the same principle. 

ThetriangleEFG = GBC + EFCB = «' + -. Now ADGrEFG:: 

P 

AG 2 : EG 2 ; or, a' + Q : a' + — : : A G 2 : E G 2 ; 

P 



whence G E = A G 






zoid, any line drawn through the middle of the first line will divide the trapezoid into 
two equivalent parts. 



DIVIDING UP LAND. 295 

2Q 

G L is obtained by taking the triangle LMG = aN ; and so for the 

P 

rest. 

Otherwise, take GE = G A \/ (-. — P^-r — ) ; and from E run 

V \m-\- n) a' 

a parallel to A D. 

If the two parts of the quadrilateral were to be equivalent, 

m = n, and we have GK = G H ju (—3- — ) ; and consequently 
G E to G A in the same ratio. 

Example. Let a quadrilateral, A B C D, be required to be thus 
divided, and let its angles, B and C, be given by rectangular co- 
ordinates, viz., A B' = 6-00 ; B' B = 9*00 ;DC' = 8'00 ;0'C = 
13-00; B'C' = 24-00. Here GH is readily found to be 29*64; 
AD G = 563*16 square chains ; and BGO = 220*16 square chains. 
Hence, by the formula, G K = 24*72 ; whence KH=GH-GK 
= 4*92 ; and the abscissas for the points E and F can be obtained 
by a simple proportion. 

The scale of the figure is 20 chains to 1 inch = 1 : 15840. 

If the quadrilateral be given by bearings, part off the desired 

area = j^fU • A B C D, by FlG 328> 

the formulas of Art. 403. ,.--'\ 

Suppose now that a quad- ,,■'' \ 

rilateral, A B C D, is to be di- 
vided into p equivalent parts, 
by lines parallel to A D. 
Measure, or calculate by trig- / 
onometry, A G. Let Q be 
the quadrilateral A B C D, and, as before, a' = B C G. Then 

aE-=-Aig-|/| g, fj" .j; GL 

" a' + Q 

GN = AG|/- p"-~y \; etc. 

( a'+Q ) 

If the quadrilateral be given by bearings, part off — . A B C D, 

2 
then part off - . ABCD, etc. ; so in any similar case. 





296 



ZAXD-S UE YEYIXG. 



444. By Lines perpendicular to a Side. Let ABCD be a 

quadrilateral which is to be divided, by 
a line perpendicular to A D, into two 
parts having a ratio = m : n. By hy- 



Fig. 329. 
E 



A G 



m 



pothesis, A B E F = — -. — . A B C D. 
m-\- n 



h 



D Taking away the triangle ABG, the 

remainder, GBEF, will be to the rest 
of the figure in a known ratio, and the position of EF. parallel 
to B G, will be found as in the last article. 



445. By Lines running in any Given Direction. To divide a 
quadrilateral A B C D into two parts : : m : n, part off from it an 



m 



area = — j— .ABCD, by the methods of Art. 407 or 408, if 
m -\-n J 

the area parted off is to be a triangle, or Art. 409 if the area parted 

off is to be a quadrilateral. 



446. By Lines starting from 
an Angle. ABCD is to be di- 
vided, by the line C E, into two 
parts having the ratio m : n. 
Since the area of the triangle 
m 



Fig. 330. 



CDE 



m-{-n 



ABCD, D E 




will be obtained by dividing this area by half of the altitude C F. 

447. By Lines starting from Points in a Side. Let it be re- 
quired to divide ABCD into two 
parts : : m : n, by a line starting 
from the point E. The area A B F E 



Fig. 331. 




is known (being 



m 



ABCD) 



m -f- n 
as also ABE; A B, B E, and E A 

being given on the ground. B E F 
will then be known = A B F E — 



BEE 



ABE. Then G F = r -g-g , and the point F is obtained by 



DIVIDING UP LAND. 



207 



Fig. 332. 



running a parallel to B E, at a perpendicular distance from it 
= GF. 

To divide a quadrilateral, ABOD, graphically, into two equiv- 
alent parts by a line from a point, 
E, on a side, proceed thus : Draw the 
diagonal C A, and from B draw a 
parallel to it, meeting D A prolonged 
in F. Mark the middle point, G, of 
FD. Join GE. From draw a 
parallel to E G, meeting D A in H. 
E H is the required line. The quad- 
rilateral could also be divided in any ratio = m : n, by dividing 
FD in that ratio. 

If the quadrilateral be given by bearings, proceed to part off the 
desired area, as in Art. 412 or 413. 




448. Let it be required to 

Fig. 333. 




F' H' are the required lines 
Let it be required to 
make the above division by 
lines starting from tiuo given 
points, P and Q. Reduce 
the quadrilateral to an equiv- 
alent triangle B E. Divide 
E B into three equal parts at 
F and G. Join Q, and, 
from G, draw G K parallel ^ 



divide a quadrilateral, A B C D, into 
three equivalent parts. 
From any angle, as C, 
draw C E, parallel to D A. 
Divide A D and E 0, each 
into three equal parts, at 
F, F', and G, G'. Draw 
B F, B F'. From G draw 
G H, parallel to F B, and 
from G' draw G' H', par- 
allel to F' B. F H and 

of division. 

Fig. 334. 




298 



LAND-SUE YEYING. 



to it. Join C P, and from F draw F L parallel to it. 
and Q K, and they will be the division-lines required. 



Join P L 



449. By Lines passing through a Point within the Figure. 

Proceed to part off the desired area as in Art. 416 or 417, accord- 
ing to the circumstances of the case. • 

Division, of Polygons. 

450. By Lines running in any Direction. Let A B C D E F G 

be a given polygon, and B H the 
direction parallel to which is to 
be drawn a line P Q, dividing 
the polygon into two parts in 
any desired ratio = m : n. The 




areaPCDEQ = 
DEFG 



m 



ABC 



m-\- n 

Taking it from the 

area B C D E H, the remainder 

will be the area B P Q H. The 

quadrilateral BCEH,CE being 

supposed to be drawn, can then be divided by the method of Art. 

443 into two parts, BPQH and P Q E C, having to each other 

a known relation. 

If D K were the given direction, at right angles to the former, 
the position of a dividing line E S could be similarly obtained. 




DIVIDING UP LAND. 



299 



451. By Lines starting from an Angle. Produce one side, A B, 
of the given polygon, both ways, and reduce the polygon to a sin- 
gle equivalent triangle, X Y Z. Then divide the base, X Y, in the 
required ratio, as at W, and draw Z W, which will be the division- 
line desired. In this figure the polygon is divided into two equiva- 
lent parts. 

If the division- line should pass outside of the polygon, as does 
Z P, through P draw a parallel to B Z, meeting the adjacent side 
of the polygon in Q, and Z Q will be the division-line desired. 



452. By Lines starting from a Point on a Side. 
414 and 415. 



See Articles 



Fig. 337. 



453. By Lines passing through a Point within the Figure. Part 
off, as in Art. 416 or 418, if a straight line be required, or by 
guess-lines and the addition of triangles, as in Art. 433, if the 
lines have merely to start from the point, such as a spring or well. 

454. Other Problems. The following is from G-ummere's " Sur- 
veying" : Question. A tract of land is bounded thus : N. 35 J° E., 
23-00 ; 1ST. 75f E., 30*50 ; S. 3J° E., 46-49 ; K 66J° W., 49-64. 
It is to be divided into four equivalent parts by two straight lines, 
one of which is to run parallel to the 

third side ; required the distance of the 
parallel division-line from the first corner, 
measured on the fourth side ; also the 
bearing of the other division-line, and its 
distance from the same corner measured 
on the first side. Ans. Distance of the 
parallel division-line from the first corner, 
32-50 ; the bearing of the other, S. 88° 
22' E. ; and its distance from the same 
corner 5*99. 

The scale of the figure is 40 chains to 1 inch = 1 : 31680. 

An indefinite number of problems on this subject might be pro- 
posed, but they would be matters of curiosity rather than of utility, 
and exercises in geometry and trigonometry rather than in survey- 




20 



300 



LAXD-SUR YEYim* 



Fig. 338. 




CHAPTER VII. 

THE PUBLIC LANDS OF THE UNITED STATES.* 

455. General System. The public lands of the United States 
of America are generally divided and laid out into squares (ap- 
proximately), the sides of which run truly north and south, or 
east and west. 

This is effected by means of meridian lines and parallels of lati- 
tude, established six miles apart. The squares thus formed are 
called Townships. They contain 36 square miles, or 23,040 acres, 
" as nearly as may be." A principal meridian, running due north 
and south, and a dase-line, running due east and west, are first 
established astronomically, and the half-mile, mile, and six-mile 
corners are permanently marked on them. These two lines form 
the basis of all the subsequent subdivision into townships and sec- 
tions. All of the lines on the public surveys, except these two and 
the standard parallels, are run with compass and chain. 

The map, Fig. 338, represents a portion of the State of 
Oregon thus laid out. The scale is 10 miles to 1 inch = 1 : 
633600. On it will be seen the " Willamette meridian," running 
truly north and south, and a "base-line," which is a "parallel of 
latitude," running truly east and west. Parallel to these, and six 
miles from them, are other lines, forming townships. All the 
townships, situated north or south of each other, form a eange. 
The ranges are named by their number east or west of the princi- 
pal meridian. In the figure are seen three ranges east and west of 

* Arts. 455 to 462 of this chapter are mainly taken from " Instructions to the Sur- 
veyor-General of Oregon, being a Manual for Field Operations," prepared, in March, 
1851, by John M. Moore, Principal Clerk of Surveys. 



302 



LAND-SUB Y EYING. 



N 



the Willamette meridian. They are noted as E. I. E., K. I. W., 
etc. The townships in each range are named by their number 
north or south of the base-line. In the figure, along the principal 
meridian, are seen four north and five south of the base-line. They 
are noted as T. 1 N., T. 2 K, T 1 S., etc.* 

Each township is divided into 36 sections, each one mile 
square, and therefore containing, "as 
nearly as maybe," 640 acres. The sec- 
tions in each township are numbered, 
as in the margin, from 1 to 36, begin- 
ning at the northeast angle of the 
township, and going west from 1 to 6, 
then east from 7 to 12, and so on alter- 
nately to section 36, which will be in 
the southeast angle of the township. 
The sections are subdivided into quar- 
ter-sections, half a mile square, and containing 160 acres, and some- 
times into half-quarter-sections of 80 acres, and quarter-quarter- 
sections of 40 acres. 

By this beautiful system, the smallest subdivision of land can 
be at once designated ; such as the northeast quarter of section 31, 
in township two south, in range two east of Willamette meridian. 



W 



6 


5 


4 


3 


2 


1 


7 


8 


9 


10 


11 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


22 


23 


24 


30 


29 


28 


27 


26 


25 


31 


32 


33 


34 


35 


36 



456. Difficulty. " The law requires that the lines of the public 
surveys shall be governed by the true meridian, and that the town- 
ships shall be six miles square — two things involving in connection 
a mathematical impossibility — for, strictly to conform to the me- 
ridian, necessarily throws the township out of square, by reason of 
the convergency of 'meridians ; hence, adhering to the true meridian 
renders it necessary to depart from the strict requirements of law 
as respects the precise area of townships, and the subdivisional 
parts thereof, the township assuming something of a trapezoidal 
form, which inequality develops itself, more and more as such, the 
higher the latitude of the surveys. In view of these circumstances, 
the law provides that the sections of a mile square shall contain 

* The marks Oi + > and a, merely refer to the dates of the surveys. They are 
sometimes used to point out lands offered for sale, or reserved, etc. 



THE PUBLIC LANDS OF THE UNITED STATES. 303 

the quantity of 640 acres, as nearly as may be; and, moreover, 
provides that, ' in all cases where the exterior lines of the town- 
ships, thus to be subdivided into sections or half-sections, shall 
exceed, or shall not exceed, six miles, the excess or deficiency shall 
be specially noted, and added to or deducted from the western or 
northern ranges of sections or half-sections in such township, ac- 
cording as the error may be in running the lines from east to west, 
or from south to north.''' 

" In order to throw the excesses or deficiencies, as the case may 
be, on the north and on the west sides of a township, according to 
law, it is necessary to survey the section-lines from south to north 
on a true meridian, leaving the result in the northern line of the 
township to be governed by the convexity of the earth and the con- 
vergency of meridians." 

Thus, suppose the land to be surveyed lies between 46° and 47° 
of north latitude. The length of a degree of longitude in latitude 
46° N. is taken as 48*0705 statute miles, and in latitude 47° N. 
as 47*1944. The difference, or convergency per square degree = 
0*8761 = 70*08 chains. The convergency per range (8 per degree 
of longitude) equals one eighth of this, or 8*76 chains; and per 
township (11J per degree of latitude) equals the above divided by 
11 J — i. e., 0*76 chain. We therefore know that the width of the 
townships along their northern line is 76 links less than on their 
southern line. The townships north of the base-line therefore be- 
come narrower and narrower than the six-mile width with which 
they start, by that amount. 

"Stakdakd Paeallels (usually called correction-lines) are 
established at stated intervals of 30 miles,* to provide for or coun- 
teract the error that otherwise would result from the convergency 
of meridians ; and, because the public surveys have to be governed 
by the true meridian, such lines serve also to arrest errors arising 
from inaccuracies in measurements. Such lines, when lying north 
of the principal base, themselves constitute a base to the surveys 
on the north of them." 

The convergency or divergency above noticed is taken up on 

* Until 1866 they were either 24 or 30 miles apart. 



30 4 LAND-SUB VEYING. 

these correction -lines, from which the townships start again with 
their proper widths. On these, therefore, there are found double 
corners, both for townships and sections, one set being the closing 
corners of the surveys ending there, and the other set being the 
standard corners for the surveys starting there. 

Auxiliaky Meridians. These are run north and south from 
the base-line, at intervals of twenty-four miles, or four townships. 

457. Running Township-Lines. " The principal meridian, the 
base-line, and the standard parallels, having been first astronomi- 
cally run, measured, and marked, according to instructions, on 
true meridians, and true parallels of latitude, the process of run- 
ning, measuring, and marking the exterior lines of townships will 
be as follows : 

Townships situated korth of the base-line and west of the 
principal meridian.* Commence at Station No. 1, being the 
southwest corner of T. 1 N. — R. 1 W., as established on the base- 
line ; thence run north, on a true meridian line, 480 chains, estab- 
lishing the mile and half-mile corners thereon, as per instructions, 
to No. 2 (the northwest corner of the same township), whereat 
establish the corner of Tps. 1 and 2 N. — Rs. 1 and 2 W. ; thence 
east, on a random or trial line, setting temporary mile and half- 
mile stakes to No. 3 (the northeast corner of the same township), 
where measure and note the distance at which the line intersects 
the eastern boundary, north or south of the true or established 
corner. Run and measure westward, on the true line (taking care 
to note all the land and water crossings, etc., as per instructions), 
to No. 4, which is identical with No. 2, establishing the mile and 
half-mile permanent corners on said line, the last half-mile of 
which will fall short of being forty chains, by about the amount of 
the calculated convergency per township, 76 links in the case above 
supposed. Should it ever happen, however, that such random line 
materially falls short, or overruns in length, or intersects the east- 
ern boundary of the township at any considerable distance from 
the true corner thereon (either of which would indicate an im- 

* The surveyor should prepare a diagram of the townships, with the numbers 
here referred to, in their proper places, as here indicated. 



THE PUBLIC LANDS OF THE UNITED STATES. 305 

portant error in the surveying), the lines must be retraced, even if 
found necessary to remeasure the meridional boundaries of the 
township (especially the western boundary), so as to discover and 
correct the error ; in doing which, the true corners must be estab- 
lished and marked, and the false ones destroyed and obliterated, to 
prevent confusion in future ; and all the facts must be distinctly 
set forth in the notes. Thence proceed in a similar manner north, 
from No. 4 to No. 5 (the N. W. corner of T. 2 N.— E. 1 W.), east 
from No. 5 to No. 6 (the N. E. corner of the same township), west 
from No. 6 to No. 7 (the same as No. 5), north from No» 7 to No. 

8 (the N. W. corner of T. 3 N., E. 1 W.), east from No. 8 to No. 

9 (the N. E. corner of the same township, and thence west to No. 

10 (the same as No. 8), or the southwest corner T. 4 N. — E. 1 W. 
Thence north, still on a true meridian line, establishing the mile 
and half-mile corners, until reaching the standard parallel or 
correction-line (which is here four townships north of the base- 
line) ; throwing the excess over, or deficiency under, four hundred 
and eighty chains, on the last half-mile, according to law, and at 
the intersection establishing the " closing corner," the distance 
of which from the standard corner must be measured and noted as 
required by the instructions. But should it ever so happen that 
some impassable barrier will have prevented or delayed the exten- 
sion of the standard parallel along and above the field of present 
survey, then the surveyor will plant, in place, the corner for the 
township, subject to correction thereafter, should such parallel be 
extended. 

Townships situated north of the .base-line, and east of the 
principal meridian. Commence at No. 1, being the southeast cor- 
ner of T. 1 N. — E. 1 E., and proceed as with townships situated 
"north and west," except that the random or trial lines will be 
run and measured west, and the trus lines east, throwing the ex- 
cess over or deficiency under four hundred and eighty chains on 
the west end of the line, as required by law ; wherefore, the sur- 
veyor will commence his measurement with the length of the de- 
ficient or excessive half-section boundary on the west of the town- 
ship, and thus the remaining measurements will all be even miles 
and half-miles. 



306 



LAND-SUB VEYING. 



458. Running Section-Lines. The interior or sectional lines of 
all townships, however situated in reference to the base and me- 
ridian lines, are laid o2 and surveyed as below : 



12 



13 



24 



25 



36 



31 



33 



34 



35 



36 





97 






71 






53 






35 




I 17 




6 






5 






4 






o 






2 




1 


99 


98 96 




72 70 




54 52 




36 34 




1816 






100 94 




95 68 




69 50 




51 3:2 




33 14 


15 


7 




8 






9 






10 






11 




12 


92 


93 91 






67 






40 






31 




13 








89 




90 65 




66147 




48 29 




30 11 


12 


18 






17 


' 


16 






15 






11 






13 


87 




86 




64 






46 






28 






10 






88 84 




85 62 




63 


44 




45 


26 




27 8 


9 


19 






20 






21 






22 






23 






21 


82 




81 






61 






43 






25. 






7 






83 


79 




80 


59 




60 


41 




42 


23 




24 5 


6 


CO 






29 






28 






27 






26 




25 


77 




76 






58 






40 






22 




4 






7S 


74 




75 


56 




57 38 




39 20 




212 


3 


31 






32 






33 




31 






35 






36 






73 






55 




|37 






19 




1 





18 



19 



30 



31 



In the above diagram, the squares and large figures represent 
sections, and the small figures at their corners are those referred to 
in the following directions : 

" Commence at No. 1 (see small figures on the diagram), the 
corner established on the township boundary for sections 1, 2, 35, 
and 36 ; thence run north on a true meridian ; at 40 chains setting 
the half-mile or quarter-section post, and at 80 chains (No. 2) 
establishing and marking the corner of sections 25, 26, 35, and 36. 
Thence east, on a random line, to No. 3, setting the temporary 
quarter-section post at 40 chains, noting the measurement to No. 
3, and the measured distance of the random's intersection north or 
south of the true or established corner of sections 25, 36, 30, and 
31, on the township boundary. Thence correct, west, on the true 
line to No. 4, setting the quarter-section post on this line exactly 



THE PUBLIC LANDS OF THE UNITED STATES. 307 

at the equidistant point, now known, between the section corners 
indicated by the small figures Nos. 3 and 4. Proceed, in like 
manner, from No. 4 to No. 5, 5 to 6, 6 to 7, and so on to No. 16, 
the corner to sections 1, 2, 11, and 12. Thence north on a ran- 
dom line, to No. 17, setting a temporary quarter-section post at 40 
chains, noting the length of the whole line, and the measured dis- 
tance of the random's intersection east or west of the true corner 
of sections 1, 2, 35, and 36^ established on the township boundary ; 
thence southwardly from the latter, on a true line, noting the 
course and distance to No. 18, the established corner to sections 
1, 2, 11, and 12, taking care to establish the quarter-section corner 
on the true line, at the distance of 40 chains from said section 
corner, so as to throw the excess or deficiency on the northern half- 
mile, according to law. Proceed in like manner through all the 
intervening tiers of sections to No. 73, the corner to sections 31, 
32, 5, and 6 ; thence north, on a true meridian line, to No. 74, 
establishing the quarter-section corner at 40 chains, and at 80 
chains the corner to sections 29, 30, 31, and 32 ; thence east, on a 
random line to No. 75, setting a temporary quarter-section post at 
40 chains, noting the measurement to No. 75, and the distance of 
the random's intersection north or south of the established corner 
of sections 28, 29, 32, and 33 ; thence west from said corner, on 
the true line, setting the quarter-section post at the equidistant 
point, to No. 76, which is identical with 74 ; thence west, on a 
random line, to No. 77, and setting a temporary quarter^ection 
post at 40 chains, noting the measurement to No. 77, and the dis- 
tance of the random's intersection with the western boundary, 
north or south of the established corner of sections 25, 36, 30, and 
31 ; and from No. 77, correct, eastward, on the true line, giving 
its course, but establishing the quarter-section post, on this line 
so as to retain the distance of 40 chains from the corner of sections 
29, 30, 31, and 32 ; thereby throwing the excess or deficiency of 
measurement on the most western half-mile. Proceed north, in a 
similar manner, from No. 78 to 79, 79 to 80, 80 to 81, and so 
on to 96, the southeast corner of section 6, where haying estab- 
lished the corner for sections 5, 6, 7, and 8, run thence, success- 
ively, on random line east to 95, north to 97, and west to 99 ; and 



308 LAND-SURVEYING. 

by reverse courses correct on true lines lack to said southeast 
corner of section 6, establishing the quarter-section corners, and 
noting the courses, distances, etc., as before described. 

"In townships contiguous to standard parallels, the above 
method will be varied as follows : In every township south of 
the principal base-line, which closes on a standard parallel, the 
surveyor will begin at the southeast corner of the township, and 
measure west on the standard, establishing thereon the mile and 
half-mile corners, and noting their distances from the pre-estab- 
lished corners. He then will proceed to subdivide, as directed 
under the above head. 

"In the townships noeth of the principal base-line, which 
close on the standard parallel, the sectional lines must be closed 
on the standard by true meridians, instead of by course-lines, as 
directed under the above head for townships otherwise situated ; 
and the connections of the closing corners with the pre-estab- 
lished standard corners are to be ascertained and noted. Such 
procedure does away with any necessity for running the randoms. 
But in case he is unable to close the lines on account of the 
standard not having been run, from some inevitable necessity, as 
heretofore mentioned, he will plant a temporary stake, or mound, 
at the end of the sixth mile, thus leaving the lines and their 
connections to be finished, and the permanent corners to be planted, 
at such time as the standard shall be extended. " 

459. Exceptional Methods. Departures from the general sys- 
tem of subdividing public lands have been authorized by law in 
certain cases, particularly on water-fronts. 

Thus, an act of Congress, March 3, 1811, authorized the sur- 
veyors of Louisiana, " in surveying and dividing such of the pub- 
lic lands in the said Territory, which are or may be authorized 
to be surveyed and divided, as are adjacent to any river, lake, 
creek, bayou, or water-course, to lay out the same into tracts, as 
far as practicable, of fifty-eight poles in front, and four hundred 
and sixty-five poles in depth, of such shape and bounded by 
such lines, as the nature of the country will render practicable 
and most convenient." Another act, of May 24, 1S24, authorizes 



THE PUBLIC LANDS OF THE UNITED STATES. 309 

lands similarly situated "to be surveyed in tracts of two acres 
in width, fronting on any river, bayou, lake, or water-course, and 
running back the depth of forty acres ; which tracts of land, 
so surveyed, shall be offered for sale entire, instead of in half- 
quarter-sections. " 

The "Instructions" from which we have quoted say: "In 
those localities where it would best subserve the interests of the 
people to have fronts on the navigable streams, and to run back 
into the uplands for quantity and timber, the principles of the 
act of May 24, 1824, may be adopted, and you are authorized to 
enlarge the quantity, so as to embrace four acres front by forty 
in depth, forming tracts of one hundred and sixty acres. But 
in so doing it is designed only to survey the lines between every 
four lots (or 640 acres), but to establish the boundary posts, or 
mounds, in front and in rear, at the distances requisite to se- 
cure the quantity of 160 acres to each lot, either rectangularly, 
when practicable, or at oblique angles, when otherwise. The 
angle is not important, so that the principle be maintained, as 
far as practicable, of making the work to square in the rear with 
the regular sectioning. 

" The numbering of all anomalous lots will commence with No. 
37, to avoid the possibility of conflict with the numbering of the 
regular sections." 

The act of September 27, 1850, authorizes the Department, 
should it deem expedient, to cause the Oregon surveys to be exe- 
cuted according to the principles of what is called the "Geodetic 
Method." 

The complete adoption of this has not been thought to be 
expedient; but "it was deemed useful to institute on the prin- 
cipal base and meridian lines of the public surveys in Oregon, 
ordered to be established by the act referred to, a system of tri- 
angulations from the recognized legal stations, to all prominent 
objects within the range of the theodolite ; by means of which 
the relative distances of such objects, in respect to those main 
lines, and also to each other, might be observed, calculated, and 
protracted, with the view of contributing to the knowledge of 
the topography of the country in advance of the progressing 



310 LAND-SURVEYING. 

linear surveys, and to obtain the elements for estimating the 
areas of valleys intervening between the spurs of the mountains." 

" Meandering" is a name given to the usual mode of survey- 
ing with the compass, particularly as applied to navigable streams. 
The "Instructions" for this are, in part, as follows: 

" Both banks of navigable rivers are to be meandered by tak- 
ing the courses and distances of their sinuosities, and the same 
are to be entered in the 'meander field-book.' At those points 
where either the township or section lines intersect the banks of 
a navigable stream, posts, or, where necessary, moulds of earth 
or stone (as noted in Art. 461), are to be established at the time 
of running these lines. These are called ' meander corners ' ; 
and in meandering you are to commence at one of those corners 
on the township-line, coursing the banks, and measuring the dis- 
tance of each course from your commencing corner to the next 
' meander corner,' upon the same or another boundary of the 
same township ; carefully noting your intersection with all inter- 
mediate meander corners. By the same method you are to mean- 
der the opposite bank of the same river. 

" The crossing distance "between the meaxdee coexees, on the 
same line, is to be ascertained by triangulation, in order that the 
river may be protracted with entire accuracy. The particulars to 
be given in the field-notes. 

"The courses and distances on meandered navigable streams 
govern the calculations wherefrom are ascertained the true areas 
of the tracts of land (sections, quarter-sections, etc.) known to 
the law as fractional, and bounding on such streams. 

"You are also to meander, in manner aforesaid, all lakes and 
deep ponds of the area of twenty-five acres and upward ; also 
navigable bayous. 

" The precise relative position of islands, in a township made 
fractional by the river in which the same are situated, is to be 
determined trigonometrically. Sighting to a flag or other fixed 
object on the island, from a special and carefully measured base- 
line, connected with the surveyed lines, on or near the river-bank, 
you are to form connection between the meander corners on the 
river to points corresponding thereto, in direct line, on the bank 



THE PUBLIC LANDS OF THE UNITED STATES. 311 

of the island, and there establish the proper meander corners, 
and calculate the distance across." 

460. Marking-Lines. "All lines on which are to be estab- 
lished the legal corner boundaries are to be marked after this 
method, viz. : Those trees which may intercept your line must 
have two chops or notches cut on each side of them, without any 
other marks whatever. These are called ' sight-trees ,' or ' line- 
trees. 9 

"A sufficient number of other trees standing nearest to your 
line, on either side of it,- are to be blazed on two sides, diag- 
onally or quartering toward the line, in order to render the line 
conspicuous, and readily to be traced, the blazes to be opposite 
each other, coinciding in direction with the line where the trees 
stand very near it, and to approach nearer each other, the farther 
the line passes from the blazed trees. Due care must ever be 
taken to have the lines so well marked as to be readily fol- 
lowed." 

461. Marking-Corners. " After a true coursing, and most exact 
measurements, the corner boundary is the consummation of the 
work, for which all the previous pains and expenditure have been 
incurred. A boundary corner, in a timbered country, is to be a 
tree, if one be found at the precise spot ; and if not, a post is 
to be planted thereat ; and the position of the corner post is to 
be indicated by trees adjacent (called bearing-trees), the angular 
bearings and distances of which from the corner are facts to be 
ascertained and registered in your field-book. 

"In a region where stone abounds, the corner boundary will 
be a small monument of stones alongside of a single marked 
stone, for a township corner — and a single stone for all other 
corners. 

"In a region where timber is not near, nor stone, the corner 
will be a mound of earth, of prescribed size, varying to suit the 
case. 

"Corners are to be fixed, for township boundaries, at intervals 
of every six miles ; for section boundaries, at intervals of every 



312 



LAXD-SUB VEYIXQ. 



mile, or 80 chains ; and, for quarter-section boundaries, at inter- 
vals of every half-mile, or 40 chains. 

" Meaxder Oorxer Posts are to be jiLanted at all those points 
where the township or section lines intersect the banks of such 
rivers, lakes, or islands, as are by law directed to be meandered, 5 ' 
as explained in Art. 459. ■ 

" When posts are used, their length and size must be propor- 
tioned to the importance of the corner, whether township, sec- 
tion, or quarter-section, the first being at least twenty-four inches 
above-ground, and three inches square. 

' ' Where a township post is a corner common to four townships, 

N 
it is to be set in the earth diagonally, thus : W^P>E, and the car- 

S 
dinal points of the compass are to be indicated thereon by a cross- 
line, or wedge (one eighth of an inch deep at least), cut or sawed 
out of its top, as in the figure. On each surface of the post is 
to be marked the number of the particular .township, and its 
range, which it faces. Thus, if the post be a common boundary 
to four townships, say one and two, south of the base-line, of 
range one, west of the meridian ; also to townships one and two, 
south of the base-line, of range two, west of the meridian — it is 
to be marked thus : 

The position of the post, which is 
here taken as an example, is shown 
in the following diagram : 

R.2W. I E. 1 TV. 

T. IS. T. 1 S. 



From N. to E. 



From X. to TV. 



From E. to S. 



From W. to S. 




30 


31 


/ 

/ 


\ 


\ 


/ 

/ 


1 6 



E. 2 W. 

T. 2 S. 



E. 1 TV. 

T. 2 S. 



"These marks are to be distinctly and neatly chiseled into the 
wood, at least the eighth of an inch deep ; and to be also marked 
with red chalk. The number of the sections which they respect- 
ively face will also oe marked on the township post. 



THE PUBLIC LANDS OF THE UNITED STATES. 313 

" Section or mile posts, being corners of sections, when they are 
common to four sections, are to be set diagonally in the earth 
(in the manner provided for township corner posts), and with a 
similar cross cut in the top, to indicate the cardinal points of the 
compass ; and on each side of the squared surfaces is to be 
marked the appropriate number of the particular one of the four 
sections, respectively, which such side faces; also on one side 
thereof are to be marked the numbers of its township and range ; 
and, to make such marks yet more conspicuous (in manner afore- 
said), a streak of red chalk is to be applied. 

ii . In the case of an isolated township, subdivided into thirty- 
six sections, there are twenty-five interior sections, the south- 
west corner boundary of each of which will be common to four 
sections. On all the extreme sides of an isolated township, the 
outer tiers of sections have corners common only to two sections 
then surveyed. The posts, however, must be planted precisely 
like the former, but presenting two vacant surfaces to receive the 
appropriate marks when the adjacent survey may be made. 

" A quarter-section or half-mile post is to have no other mark 
on it than J S., to indicate what it stands for. 

"Township corner posts are to be- notched with six notches 
on each of the four angles of the squared part set to the cardi- 
nal points. 

"All mile-posts on toivnship lines must have as many notches 
on them, on two opposite angles thereof, as they are miles dis- 
tant from the township corners, respectively. Each of the posts 
at the corners of sections in the interior of a township must in- 
dicate, by a number of notches on each of its four corners di- 
rected to the cardinal points, the corresponding number of miles 
that it stands from the outlines of the township. The four sides 
of the post will indicate the number of the section they respect- 
ively face. Should a tree be found at the place of any corner, 
it will be marked and notched, as aforesaid, and answer for the 
corner in lieu of a post ; the kind of tree and its diameter being 
given in the field-notes. 

"The position of all corner posts, or corner trees of what- 
ever description, which may be established, is to be perpetuated 



314- LAND-SURVEYING. 

in the following manner, viz. : From such post or tree the 
courses shall be taken, and the distances measured, to two or 
more adjacent trees, in opposite directions, as nearly as may be, 
which are called ' bearing-trees, ,' and are to be blazed near the 
ground, with a large blaze facing the post, and haying one notch 
in it, neatly and plainly made with an axe, square across, and 
a little below the middle of the blaze.. The kind of tree and 
the diameter of each are facts to be distinctly set forth in the 
field-book. 

" On each bearing- tree the letters B. T. must be distinctly cut 
into the wood, in the blaze, a little above the notch, or on the 
bark, with the number of the range, township, and section. 

"At all township corners, and at all section corners, on range 
or township lines, four bearing- trees are to be marked in this 
manner, one in each of the adjoining sections. 

"At interior section corners four trees, one to stand within 
each of the four sections to which such corner is common, are 
to be marked in the manner aforesaid, if such be found. 

"From quarter-section and meander corners two bearing-trees 
are to be marked, one within each of the adjoining sections. 

" Stones at township corners (a small monument of stones be- 
ing alongside thereof) must have six notches cut with a pick or 
chisel on each edge or side toward the cardinal points ; and 
where used as section corners on the range and township lines, 
or as section corners in the interior of a township, they will also 
be notched by a pick or chisel, to correspond with the directions 
given for notching posts similarly situated. 

"Stones, when used as quarter-section corners, will have J- cut 
on them ; on the west side on north and south lines, and on the 
north side on east and west lines. 

"Whenever bearing-trees are not found, moulds of earth, or 
stone, are to be raised around posts on which the corners are to 
be marked in the manner aforesaid. Wherever a mound of earth 
is adopted, the same will present a conical shape ; but at its base, 
on the earth's surface, a quadrangular trench will be dug : a 
spade-deep of earth being thrown up from the four sides of the 
line, outside the trench, so as to form a continuous elevation along 



TEE PUBLIC LANDS OF THE UNITED STATES. 315 

its outer edge. In mounds of earth, common to four townships 
or to four sections, they will present the angles of the quadrangu- 
lar trench (diagonally) toward the cardinal points. In mounds 
common only to tivo townships or tivo sections, the sides of the 
quadrangular trench will face the cardinal points. 

" Prior to piling up the 'earth to construct a mound, in a cavity 
formed at the corner boundary point is to be deposited a stone, 
or a portion of charcoal, or a charred stake is to be driven twelve 
inches down into such center point, to be a witness for the fu- 
ture. 

"The surveyor is further specially enjoined to plant, midway 
between each pit and the trench, seeds of some tree, those of fruit- 
trees adapted to the climate being always to be preferred. 

"Double corners are to be found nowhere except on the 
standard parallels or correction-lines, whereon are to appear both 
the corners which mark the intersection of the lines which close 
thereon, and those from which the surveys start in the opposite 
direction. 

" The corners which are established on the standard parallel, at 
the time of running it, are to be known as i Standard Corners,'' 
and, in addition to all the ordinary marks (as herein prescribed), 
they will be marked with the letters S. C. The ' closing corners ' 
will be marked C. C." 

462. Field-Books. There should be several distinct and sepa- 
rate field-books, viz. : 

" 1. Field-notes of the meridian- and base lines, showing the 
establishment of the township, section, or mile, and quarter-section or 
half-mile, boundary corners thereon ; with the crossings of streams, 
ravines, hills, and mountains ; character of soil, timber, minerals, 
etc. These notes will be arranged, in series, by mile-stations, from 
number one to number . 

" 2. Field-notes of the ' standard parallels, or correction- 
lines/ showing the establishment of the township, section, and 
quarter-section corners, besides exhibiting the topography of the 
country on line, as required on the base and meridian lines. 

" 3. Field-notes of the exterior lines of townships, showing 

21 



316 LAND-SUR V EYING. 

the establishment of the corners on line, and the topography, as 
aforesaid. 

" 4. Field-notes of the subdivisions of townships into sec- 
tions and quarter-sections ; at the close whereof will follow the 
notes of the meanders of navigable streams. These notes will also 
show, by ocular observation, the estimated rise and fall of the land 
on the line. A description of the timber, undergrowth, surface, 
soil, and minerals, upon each section-line, is to follow the notes 
thereof, and not to be mixed up with them." 

5. The " Geodetic Field-Book," comprising all triangulations. 
angles of elevation and depression, leveling, etc. 

The examples on the next two pages, taken from the " Instruc- 
tions " which we have followed throughout, will show what is re- 
quired. 

The ascents and descents are recorded in the right-hand col- 
umns. 

For full details of public-land surveying, see " System of Eec- 
tangular Surveying," by J. H. Hawes. 

" Instructions " are issued from the General Land-Office from 
time to time, giving any changes in methods of work, or of mark- 
ing-points. 



TEE PUBLIC LANDS OF THE UNITED STATES. 317 



FIELD-NOTES OF 

THE EXTERIOR LINES 

OF AN ISOLATED TOWNSHIP. 



Field-notes of the Survey of Township 25 north, of Range 2 west, of the Willamette 
meridian, in the Territory of Oregon, by Robert Acres, Deputy-Surveyor, under his 
contract No. 1, bearing date the 2d day of January, 1851. 



£ 


Chs. Iks. 




Feet. 






Township lines commenced January 20, 1851. 




n 




Southern boundary variation 18° 41' E. 




o 


East. 


On a random line on the south boundaries of sections 31, 32, 33, 




w 




34, 35, and 36. Set temporary mile and half-mile posts, and 




$£, 




intersected the eastern boundary 2 chains 20 links north of 




-^ 




the true corner 5 miles 74 chains 53 links. 




S 
o 




Therefore the correction will be 5 chains 47 links W., 37'1 links 




g 




S. per mile. 




Ch 


True southern boundary variation 18° 41' E. 






West. 


On the southern boundary of sec. 36, Jan. 24, 1851. 






40-00 


Set qr. sec. post from which 


a 10 






a beech 24 in. dia. bears N. 11 E. 3S links dist. 




a 




a do. 9 do. do. S. 9 E. 17 do. 




© 


62-50 


a brook 8 1. wide, course N. W. . 


d 10 


r& 


80-00 


Set post cor. of sees. 35 & 36, 1 & 2, from which 


a 5 






a beech 9 in. dia. bears S. 46 E. 8 1. dist. 








a do. 8 do. do. S. 62 W. 7 do. 




B 




a w. oak 10 do. do. N. 19 W. 14 do. 




£ 




a b. oak 14 do. do. N. 29 E. 16 do. 
Land level, part wet and swampy ; timber, beech, oak, ash, 






West. 


hickory, etc. 






On the S. boundary of sec. 35 — 




P 


40-00 


Set qr. sec. post, with trench, from which 


a 10 


Pi 
O 




a beech 6 in. dia. bears N. 80 E. 8 1. dist. 




'O 




planted S. W. a yellow-locust seed. 




<U 


6500 
8000 


To beginning of hill 


a 5 


03 


Set post, with trench, cor. of sees. 34 & 35, 2 & 3, from which 


«20 


s 




a beech 10 in. dia. bears S. 51 E. 13 1. dist. 




"-J3 




a do. 10 do. do. N. 56 W. 9 do. 








Planted S. W. a white-oak acorn, 








N. E. a beechnut. 




a 

O 


West. 


Land level, rich, and good for farming; timber same. 






On the S. boundary of sec. 34 — 




03 

j5 


40-00 


Set qr. sec. post, with trench, from which 


a 5 


O 




a black oak 10 in. dia. bears N. 2 E. 635 1. dist. 




« 




Planted S. W. a beechnut. 




o 


80-00 


To corner of sections 33, 34, 3 and 4, drove charred stakes ; 


a 10 


"-S 




raised mound, with trench, as per instructions, and 




to 




Planted N. E. a white-oak acorn ; N. W. a yellow-locust seed ; 




P 




S. E. a butternut ; S. W. a beechnut. 




3 
O 




Land level, rich, and good for farming ; some scattering oak and 




S 




walnut. 






Etc., etc., etc. 





318 



LAND-SUR V EYING. 



FIELD-X0TE3 OF THE 

SUBDIVISIONAL OR SECTIONAL LINES, 

AXD MEANDERS. 



Township 25 A 7 ., Range 2 W., Willamette Mer. 





Chs. Iks. 

North. 

9-19 

29-97 

40-00 

51-90 

70-73 
80-00 

East. 
9-00 
15-00 
40-00 
55-00 
72-00 
80-00 


Subdivisions. Commenced February 1, 1851. 

Between sees. 35 and 36 — 

A beech 30 in. dia 


Feet. 
d 10 




A beech 30 in. dia 


d 5 


a5 

d 

- 


Set qr. sec. post, from which 

a beech 15 in. dia. bears S. 48 E. 12 1. dist. 

a do. 8 do. do. N. 23 \V '. 45 do. 
A beech 18 in. dia 


d 5 
d 5 


H 


A sugar 30 in. dia 


d 8 




Set a post cor. of sees. 25, 26, 35, 36, from which 

a beech 24 in. dia. bears X. 62 W. 17 1. dist. 

a poplar 36 do. do. S. 66 E. 34 do. 

a do. 20 do. do. S. 70 TV. 50 do. 

a beech 28 do. do. N. 60 E. 45 do. 
Land level, second rate ; timber, beech, poplar, sugar, and 

und'gr. spice, etc. 


d 2 


5 
o 
-a 

5 


On random line between sees. 25 and 36 — 

A brook 30 1. wide, course N 

To foot of hill 


d\0 

d 10 


Set temporary qr. sec. post 

To opposite foot of bill 


a 60 
d40 




A brook 15 1. wide, course N 

Intersect E. boundarv at post 


<7 20 
a 10 




Land level, second rate ; timber, beech, oak, ash, etc. 






Etc., etc., etc. 





Meaxdees of Chickeeles River. 

Beginning at a meander post in the northern township boundary, and thence on the 
left bank down-stream. Commenced February 11, 1851. 



Courses. 



Distances. 
Chs. Iks. 



S. 76 W. 
S. 61 W. 
S. 61 W. 

S. 54 TV. 
S. 40 TV. 
S. 50 W. 
S. 37 TV. 
S. 44 TV. 
S. 36 TV. 



18-46 

10-00 

8-18 



10-69 
5-59 
8-46 
16-50 
21-96 
2753 



Eemabks. 



In section 4 bearing to corner sec. 4 on right bank N. 70' W. 
Bearing to cor. sec. 4 and 5, right bank X. 52" TV. 
To post in line between sections 4 and 5, breadth of river by 
triangulation 9 chains 51 links. 



In section 5. 



To upper corner of John Smith's claim, course E. 

To post in line between sections 5 and 8, breadth of river by 

triangulation S chains 78 links. 



Etc., 



etc., 



THE SOLAR COMPASS. 319 

THE SOLAR COMPASS. 

463. Nearly all of the lines required in the public-land surveys 
are meridians and parallels of latitude. Meridians may be located 
by the methods given in 8 Chapter III, but the easiest method is 
with the Solar Compass. 

There are several varieties of this instrument, all of which are 
constructed on the same principle, and are modifications of the 
instrument invented by William A. Burt, and patented by him in 
1836. 

Before describing the solar compass, it will be necessary to de- 
fine the terms to be used. 

464. Definitions. The axis of the earth is the imaginary line about 
which it revolves. The points in which the axis meets the surface of the 
earth are called the poles of the earth. 

Meridians are great circles of the earth's surface, passing through the 
poles. The equator is a great circle of the earth's surface, 90° from the 
poles. Parallels of latitude are small circles of the earth's surface parallel 
to the equator. Latitude is the distance north or south from the equator, and 
is measured on a meridian circle. Longitude is distance east or west from 
some established meridian. The meridian of Greenwich, England, is usually 
taken as the prime meridian, from which longitude is reckoned. 

Astronomical Terms. Conceive all of the heavenly bodies projected upon 
the concave surface of a sphere, of which the earth is the center, and whose 
radius is infinitely great when compared with that of the earth. This is 
called the Celestial Sphere. 

If the axis of the earth be prolonged, the points in which it meets the 
celestial sphere are called the north and south poles of the heavens, and the 
line joining them is called the axis of the celestial sphere. The apparent 
revolution of the heavenly bodies about the axis of the celestial sphere is due 
to the rotation of the earth on its axis once in twenty-four hours, 

A plane passed tangent to the earth at the feet of an observer is the sen- 
sible horizon ; and a plane passed, parallel to this, through the center of the 
earth, is the rational horizon. Since the radius of the earth is infinitely small 
in comparison with that of the celestial sphere, if the planes of the rational 
horizon and sensible horizon he extended in every direction indefinitely, they 
will meet the celestial sphere in one great circle, called the celestial horizon. 
If the plane of the earth's equator be extended indefinitely, it will meet the 
celestial sphere in a great circle, called the celestial equator, or equinoctial. 

If through any place a line be passed, perpendicular to the plane of the 
horizon, the point in which it meets the celestial sphere above the observer 
is called the zenith; and the point in which it meets the celestial sphere be- 
low the observer, the nadir. 



320 LAND-SURVEYING. 

Great circles passing through the zenith and nadir are vertical circles. 

The zenith distance of a heavenly body is its angular distance from the 
zenith, and is measured on a vertical circle. The altitude of a body is its 
angular distance above the celestial horizon, and is measured on a vertical 
circle. Altitude and zenith distance are complements of each other. 

Great circles passing through the poles of the celestial sphere are called 
circles of declination, or hour-circles. The declination of a heavenly body is 
its angular distance north or south from the equinoctial, and is measured on 
a circle of declination. 

The celestial meridian of any place is a great circle passing through the 
zenith, and through the poles of the celestial sphere. The line in which the 
plane of the celestial meridian meets the plane of the horizon is the terres- 
trial meridian, or true north and south line. 

The hour-angle of a heavenly body is the angle at the pole between the 
meridian and the declination circle passing through the body. 

The parallactic angle is the angle at the body between the declination 
circle and vertical passing through the body. 

The azimuth of a heavenly body is the angle between the celestial me- 
ridian and a vertical circle passing through the body, and is measured on the 
celestial horizon. 

If an observer be at the equator, the celestial horizon will pass through 
the poles of the heavens, and the celestial equator through the zenith. For 
each degree which the observer travels northward on the earth, the north 
pole of the heavens will appear to rise one degree above the horizon, and 
the celestial equator will appear to move one degree southward from the 
zenith. The latitude of a place, then, is equal to the altitude of the elevated 
pole, or to the declination of the zenith. In the northern hemisphere the 
north pole of the heavens is the elevated pole. 

The earth revolves around the sun in an elliptical orbit once in a year. This 
gives the sun an apparent motion around the earth. The path of the earth, 
or the apparent path of the sun in the heavens, is called the ecliptic. It is a 
great circle on the celestial sphere, making an angle with the celestial equa- 
tor of about 23° 27'. The two points in which the ecliptic meets the equi- 
noctial are called the equinoxes. The sun is on the equinoctial the 21st of 
March. This is the vernal equinox. It then moves north of the equator, 
increasing constantly in northern declination, until the 21st of June, when 
its declination is about 23° 27' north. This is the northern summer solstice. It 
then decreases in declination until September 21st, when it is again on the 
equinoctial. This is the autumnal equinox. It then moves south of the 
equator, increasing in southern declination until December 21st, when its 
declination is about 23° 27' south. This is the northern winter solstice. It 
then decreases in declination until March 21st, when it again arrives at the 
vernal equinox. The declination of the sun is given in the " Xautical Alma- 
nac " for every day in the year. 

The transit of a heavenly body is its passage across the celestial me- 
ridian. 

A sidereal day is the interval of time between two successive transits of 



THE SOLAR COMPASS. 



321 



the vernal equinox. A solar day is the interval of time between two suc- 
cessive transits of the sun. The apparent motion of the sun is not uniform, 
and hence use is made of a fictitious, or mean sun, moving on the equinoctial 
with a uniform motion, and keeping mean solar time. This is the time kept 
by clocks and watches. The time indicated by the true sun is called appar- 
ent solar time. This is tlie time given by sun-dials. The difference between 
apparent solar time and mean solar time is called the equation of time. The 
equation of time is zero four times in a year, and its maximum value is about 
sixteen minutes. It is given in the " Nautical Almanac" for every day in the 
year. 

A ray of light, passing from a rarer to a denser medium, is bent, or re- 
fracted, toward a perpendicular to the surface of the second medium at the 
point where the ray enters. The atmosphere surrounding the earth varies 
in density, being denser as we approach the surface of the earth. The light 
coming from a heavenly body, and passing through the atmosphere, will be 
constantly bent toward a perpendicular to the surface of the earth, and its 
path will be a curve, and not a straight line. The apparent direction of a 
heavenly body will be tan- 
gent to this curve where it 
meets the eye of the ob- 
server. The difference be- 
tween the apparent and 
the true positions of a 
heavenly body is called re- 
fraction. It is zero at the 
zenith, and about 33' at 
the horizon ; 45° from the 
zenith it is about 57". 

Refraction increases the 
altitude of a heavenly body 
and decreases the zenith 
distance. 

In Fig. 339, N S repre- 
sents the axis of the celes- 
tial sphere, N the north 
pole, and S the south pole. 
E D Q is the equinoctial, 
H A O the horizon, and 

H Z O X the meridian. Z A X is a vertical circle, N D S a declination-circle. 
(the position of the earth) is the center of the celestial sphere. Z is the 
zenith and X the nadir. Let P be any point on the celestial sphere. A P is 
its altitude, PZ its zenith distance, and PD its declination; Z N P its hour- 
angle, Z P N its parallactic angle, and NZP its azimuth. 




465. The solar compass differs from the ordinary compass, Fig. 
135, in having a solar apparatus, instead of a magnetic needle, for 



determining the meridian. 



322 



LAND-SURVEYING. 



In the figure, a is the latitude-arc, whose center of motion is in 
two pivots, one of which is shown at d. It is furnished with a 
clamp, slow-motion screw,/, and vernier, e. 

The declination-arc is shown at b. The movable arm, h, has 
its center of motion in a pivot at g, and is furnished with a clamp, 
vernier, v, and a slow-motion screw, Tc. 





4, 



■^M 



c .;:;^\ i 



i 



I <^*S 




:^' . 







The plane of the hour-arc, c, is at right angles to the latitude- 
arc, and its center is in the polar axis p. 

The declination-arc and latitude-arc are read to minutes by the 
verniers. The hour-arc is o T aduated to half-degrees, and is figured 



both for hours and degrees. 



1* 




m?^. 



THE SOLAR COMPASS. 323 

Attached to each end of the arm h is a rectangular block of 
brass, in which is set a convex lens, whose focus is on a silver 
plate attached to the face of the opposite block. The silver plate 
is marked by two sets of parallel lines, at right angles to each 
other, as shown in Fig. 341 ; b b are called the hour-lines, and c c 
the equatorial lines. The distance between the 

"rip QAl 

hour-lines and between the equatorial lines is 
equal to the diameter of the image of the sun, 
formed by the lens in the opposite block. 

The needle-box n contains a magnetic nee- 
dle, and is furnished with an arc of about 36° in extent, graduated 
to half-degrees. The needle-box can be moved about its center by 
the slow-motion screw t. 

The sight and levels are similar to those of the ordinary compass. 

The equatorial sights, u and n, attached to the upper side of 
the rectangular lens-blocks, are used in the adjustments. 

The adjuster, also used in adjusting the instrument, is kept in 
the instrument-box, and is not shown in the figure. 

The compass-sights are attached to the lower plate, and the 
solar apparatus, levels, and needle-box to the upper plate. The 
horizontal limb is read to single minutes by the vernier. 

Suppose the instrument to be set up and leveled, with the lati- 
tude-arc toward the south. If, now, the latitude-arc be set to the 
latitude of the place of observation (that is, so that the plane of 
the hour-arc makes an angle with the vertical equal to the latitude 
of the place), the plane of the hour-arc will then be in the plane of 
the celestial equator, and the polar axis will be parallel to the axis 
of the earth, and will point toward the north pole of the heavens. 
If the sun be on the celestial equator, the declination-arm, h, may 
be set at zero on the declination-arc, and it will then lie in the 
plane in which the sun appears to move. If the declination-arc 
be turned so as to point toward the sun, the lens in the block to- 
ward the sun will form an image on the silver plate attached to the 
opposite block. By means of the polar axis, p, the declination- 
arm may be turned so as- to follow the sun all day. 

When the sun is not at the equinoxes, set off its declination on 
the declination-arc, and the declination-arm, when turned about on 



324 LAND-SUR VEY1N0. 

the axis, p, will still turn in the plane in which the sun appears to 
move. When the sun is in south declination, turn the declination - 
arc away from the sun ; and when the sun is in north declination, 
turn the declination-arc toward the sun. 

When the instrument is in perfect adjustment, and is properly 
set up and leveled, the image of the sun can not be brought be- 
tween the equatorial lines, unless the sights are in the plane of the 

meridian. 

Adjustments. 

466. The adjustments will be given in the order in which they 
should be made. In describing each adjustment, it will be sup- 
posed that the instrument has been properly set up and leveled, 
and the latitude-arc turned toward the south. 

467. First Adjustment. To cause the level-bullies to remain in 
the center of the tubes when the instrument is turned around on its 
vertical axis. The verification and rectification are the same as 
those given for the common compass. 

468. Second Adjustment. To adjust the equatorial lines and 
solar lenses. Detach the declination-arm, h, by removing the neces- 
sary screws, and attach in its place the adjuster, replacing the 
screws of the pivot, and also of the clamp. 

Place the arm h on the adjuster, with the same side against the 
declination-arc as before it was detached. Then, by means of the 
vertical axis of the instrument, the declination and latitude arcs, 
and the leveling-screws, turn the arm in the direction of the sun, 
and bring the image of the sun between the equatorial lines. Then 
turn the arm half over, bringing the opposite faces of the blocks in 
contact with the adjuster. 

If the sun's image remains between the equatorial lines, the sil- 
ver plate is in its proper position. If not, loosen the screws which 
hold the plate, and move the plate so as to correct half of the ap-. 
parent error. Verify the work by repeating the above operation, 
until the image remains between the lines in both positions of the 
arm. 

To adjust the other plate, turn the arm end for end on the ad- 
juster, and then proceed as for the first plate. 



THE SOLAR COMPASS. 325 

When both plates have been properly adjusted, remove the ad- 
juster, and replace the declination-arm and its attachments. 

469. Third Adjustment. To adjust the vernier of the declina- 
tion-arc. Set the vernier of the declination-arc at zero. Turn the 
declination-arm h so as to point toward the sun. Bring the sun's 
image between the equatorial lines, by means of the slow-motion 
screw of the latitude-arc and the parallel plate-screws, as in the 
second adjustment. Then revolve the arm so as to bring the op- 
posite solar lens toward the sun. If the sun's image now comes 
between the equatorial lines, no adjustment is necessary. If not, 
correct half of the apparent error by means of the slow-motion 
screw h. Verify the work by repeating the above operation until 
the image comes between the lines in both positions of the arm. 
The zero of the vernier will now not coincide with the zero of the 
arc. Make it do so by loosening the screws which, hold the ver- 
nier, and moving the vernier. 

470. Fourth Adjustment. To adjust the Solar Apparatus to 
the Compass-Sights. Set the vernier of the horizontal limb at zero. 
Raise the latitude-arc until the polar axis is horizontal, and set the 
vernier of the declination-arc at zero. Direct the equatorial sights 
at some distant point. If the same point is seen through the sights, 
no adjustment is necessary. If not, the sights must be changed, 
or some equivalent adjustment made, which can only be done by 
an instrument-maker. 

Field -Work. 

471. Before the instrument can be used in the field, it is neces- 
sary to determine what angles are to be set off on the declination- 
arc and on the latitude-arc. 

On the declination-arc, both the declination of the sun and the 
correction for refraction must be provided for. 

472. Declination. The declination of the sun at noon at Green- 
wich, England, is given in the "Nautical Almanac" for every day 
in the year, together with the hourly change in declination. 

To determine the declination at any place for any time, a cor- 
rection will need to be applied for difference of declination due to 



326 LAND-SURVEYING. 

the difference of time corresponding to difference of longitude, and 
also for change of declination for different hours of the day. 

For example, suppose we wish to find the declination of the sun 
at Schenectady, New York, for the different hours of the day on 
May 1, 1885. The longitude of Schenectady is 73° 55' 50" west. 
This in time is 4 h. 55 m. 43 sec, or approximately (and near enough 
for this purpose) 5 hours. From the " Nautical Almanac" we find 
that the declination of the sun at Greenwich, noon on May 1st, to 
be 15° 12' 37-5" north, and the hourly difference is 45". 

When it is noon at Greenwich, it is 7 o'clock in the morning at 
Schenectady, and at that time the declination of the sun is 15° 
12' 37". 

For the successive hours of the day we have only to add the 
hourly difference in declination, 55'' (the sun at that time having a 
motion northward from the equator). 

473. Refraction. Tables of refraction have been calculated, 
giving the amount of refraction for different altitudes from the 
horizon. These tables, however, give the refraction in a vertical 
plane, and are not directly applicable for use as a correction in 
declination. It is evident that, in revolving the declination-arc 
around the polar axis, the declination-arc will not lie in the plane 
of a vertical circle, except when it is placed in the plane of the me- 
ridian. The correction for refraction, to be set off on the declina- 
tion-arc, will not, therefore, be equal to the refraction given in the 
tables except at noon. 

The proper correction for refraction to be set off on the decli- 
nation-arc varies with the latitude, declination of the sun, and 
hour-angle of the sun. 

From Chauvenet's "Astronomy," Art. 120, we have : 
Befraction in declination = Jc' . tan. z . cos. q. 

The value of Id may be taken from Table II, Chauvenet's "As- 
tronomy." Its mean value is about 57", and this may be employed 
when very precise results are not required. 

z is the zenith distance, and q the parallactic angle. 

From Art. 15, Chauvenet's "Astronomy," we have : 
tan. z . cos. q = cot. (8 + N), 



THE SOLAR COMPASS. 327 

in which 8 = declination of the sun, and N is an auxiliary quan- 
tity. Tan. N equals cot. <f> . cos. t, in which <f> is the latitude of 
the place, and t the hour-angle of the sun. 

The tables of Eefraction in Declination * are calculated by the 
above formulas. 

In the tables the hour-angle denotes the distance of the sun 
from the meridian in hours. Thus, at 7 o'clock A. m. the value of 
the hour-angle is five hours. The north declinations are indicated 
by + and the south declinations by — . 

When the sun is in north declination, the refraction in declina- 
tion given by the tables is additive. When the sun is in south 
declination, it is subtractive. 

No tables of refraction can be relied upon for altitudes of less 
than five degrees. 

To use the tables, suppose the declination, corrected for re- 
fraction, be required for each hour of the day, May 1, 1885, at 
Schenectady, New York. 

By Art. 472 we found that the declination at 7 o'clock in the 
morning was 15° 12' 37". The latitude of Schenectady is 42° 49'. 
(Take tabular values for 42° 30'.) 

In the tables we find that the refraction in declination for lati- 
tude 42° 30', when the sun's declination is 15°, and hour-angle 5 
hours, is 1' 36". Adding this to 15° 12' 37", we have 15° 14' to be 
set off on the declination-arc. 

474. To determine the Latitude. Set off on the declination-arc 
the declination of the sun at noon on the given day (corrected for 
refraction). 

A few minutes before noon, set up and level the instrument, set 
the declination-arc at 12 o'clock on the hour-arc, and turn the in- 
strument horizontally until the declination -arm is directed toward 
the sun. Move the latitude-arc vertically so as to bring the sun's 
image between the equatorial lines. As the sun moves toward the 
meridian, turn the instrument horizontally so as to keep the image 
between the hour-lines, and move the latitude-arc so as to keep the 

* These tables were calculated by Edward W. Arms, C. E., for W. & L. E. 
Gurley. 



328 LAND-SURVEYING. 

image between the equatorial lines. So long as the sun is ascend- 
ing, the image will move downward on the plate. When the sun 
has passed the meridian, and begins to descend, the image will 
move upward. When the image begins to move upward, the 
reading on the latitude-arc will give the latitude of the place. 

475. To determine the "Meridian," or true North and South 
Line, Set off on the latitude-arc the latitude of the place, and on 
the declination -arc the declination of the sun at the time, corrected 
for refraction. Level the instrument, clamp the horizontal plates 
at zero, turn the latitude-arc approximately south, and direct the 
declination-arm toward the sun. Then with one hand turn the 
instrument horizontally, and with the other revolve the declination- 
arm on the polar axis, until the image of the sun is brought between 
the equatorial lines. The sights will then point north and south. 

476. Running Lines. The meridian being given by the solar 
compass, it can be used for determining the bearing of lines in the 
same way as an ordinary compass, but with greater precision, as 
the meridian is more accurately determined, and the angles are 
read by the vernier to single minutes. 

477. Use of the Magnetic Needle. Since the solar compass gives 
the true meridian, and the magnetic needle the " magnetic merid- 
ian," the declination of the magnetic needle can be read off directly 
from the magnetic needle. If the needle be kept at zero of the 
compass-box arc, by turning the box with its tangent-screw, the 
declination of the needle can be read to minutes on the arc which 
shows the movement of the compass-box. 

By constantly noting the declination of the needle, or by mov- 
ing the needle-box so as to keep the needle reading zero, lines may 
be run by the needle, while the sun is obscured, or at such times as 
for any reason the solar apparatus is not reliable, as when the sun 
is near the horizon or the meridian. 

478. Solar Attachment.* The solar apparatus may be attached 
to a transit, as shown in Fig. 342. 

* This attachment, shown in Fig. 842, is manufactured by W. & L. E. Gurley, 
Troy, New York. 



THE SOLAR COMPASS. 



329 



The " polar axis " of the solar apparatus is attached to the hori- 
zontal axis of the telescope, and projects upward. The " hour- 
circle " is the small graduated circle, shown above the telescope. 

Fig. 342. 




Engineer's Transit, with Solar Attachment. 



On the "polar axis" rests the frame, which carries the "declina- 
tion-arc," and the "arm" with its slow-motion attachments, "solar 
lenses," and "equatorial lines," as before described. 



330 LAND-SURVEYING. 

The vertical circle, or arc, of the transit, is used for a "lati- 
tude-arc." 

Adjustments. 

479. The first, second, and third adjustments are similar to 
those of the solar compass, already explained. 

480. To adjust the Polar Axis. Level the instrument carefully, 
and then level the telescope by means of the level attached to it. 
Set the arm of the declination-arc at zero, and bring it parallel to 
the telescope. Place an adjusting level, shown in Fig. 343, on the 

Fig. 343 




rectangular blocks attached to the declination-arm. If the bubble 
remains in the center, the polar axis needs no adjustment in the 
plane of the axis of the telescope. If not, bring the bubble to the 
center by means of the two capstan-head screws under the hour- 
circle, and in line with the telescope. Then turn the declination- 
arm on the polar axis until it is parallel to the telescope axis, and 
at right angles to its former position. If the bubble now remains 
in the center, no adjustment is necessary. If not. bring the bubble 
to the center by means of the pair of capstan-head screws under the 
hour-circle and in line with the telescope axis. Verify, and repeat 
the above operations until the bubble of the adjusting level will 
remain in the center while the declination-arm is revolved horizon- 
tally on the polar axis. 

481. To adjust the Hour- Arc. When the telescope is 'in the 
plane of the meridian, the index of the hour-circle should give ap- 
parent solar time — that is, mean solar time ± the equation of time. 
If the index does not point to the proper division, it can be made 
to do so by loosening the screws on the top of the hour-circle, and 
turning it until the correct time is indicated by the index. 



THE SOLAR COMPASS. 
Fig. 344. 



331 




22 



Transit, with Solar Attachment. 



332 LAND-SUR VEYING. 

482. The method of using the solar apparatus on the transit is 
so nearly the same as that on the compass, already given, that no 
separate directions will be necessary. 

483. Fig. 344 represents a transit with another form of solar 
attachment.* It consists essentially of a small telescope and level, 
the telescope being mounted in. standards, in which it can be ele- 
vated or depressed. The standard revolves around an axis, called 
the polar axis, which is fastened to the telescope axis of the transit 
instrument. The telescope, called the " solar telescope," can thus 
be moved in altitude and azimuth. It is provided with shade- 
glasses to subdue the glare of the sun, as well as a prism to observe 
with greater ease when the declination is far north. Two pointers 
attached to the telescope to approximately set the instrument are 
so adjusted that when the shadow of the one is thrown on the other 
the sun will appear in the field of view. 

Adjustment of the Apparatus. 

First. Attach the " polar axis" to the main telescope axis in 
the center at right angles to the line of collimation. The base of 
this axis is provided with three adjusting-screws for this purpose ; 
by means of the level on the solar telescope this condition can be 
readily and accurately tested. 

Second. Point the transit telescope — which instrument we as- 
sume to be in adjustment — exactly horizontal, and bisect any dis- 
tant object. The transit level will then be in the middle of the 
scale. Point the "solar telescope" also horizontally by observing 
the same object, and adjust its level to read zero, for which pur- 
pose the usual adjusting-screws are provided. 

DlEECTIONS FOE USING THE ATTACHMENT. 

First. Take the declination of the sun as given in the "Nauti- 
cal Almanac " for the given day and hour, and correct it for refrac- 
tion and hourly change. Incline the transit telescope until this 
amount is indicated by its vertical arc. If the declination of the 

* Invented by G. N. Saegm tiller, and manufactured by Fauth & Co., Washington, 
D. C, from whose catalogue the description is taken. 



THE SOLAR COMPASS. 333 

sun is north, depress it ; if south, elevate it. Without disturbing 
the position of the transit telescope, bring the solar telescope to 
a horizontal position by means of its level. The two telescopes 
will now form an angle which equals the amount of the declina- 
tion. 

Second. Without disturbing the relative positions of the two 
telescopes, incline them and set the vernier to the latitude of the 
place. 

The vertical axis of the " solar attachment " will then point to 
the pole, the apparatus being in fact a small equatorial. 

By moving the transit and the " solar attachment " around 
their respective vertical axes, the image of the sun will be brought 
into the field of the solar telescope, and after actually bisecting it 
the transit telescope must be in the meridian, and the compass- 
needle indicates its deviation at that place. 

To locate a Parallel of Latitude. 
481 In Fig. 345, let P be the pole of the earth, PA and PB 
the meridians, and A B the desired parallel. 

First Method. If from A a line, A C, 
be run perpendicular to the meridian A P, 
it is evident that, owing to the convergence 
of the meridians, the perpendicular will not 
coincide with the parallel of latitude through 
A. In north latitudes, as in the United 
States, the perpendicular, A C, will run to 
the south of the parallel, A B. 

To find the distance C B, when the lati- 
tude of the starting-point A, and the distance A C are known. 
In the triangle PAC, right-angled at A : 
cos. P C = cos. A P x cos. A C. 

BC = PC-PB, and A P = B P = co-latitude. 

.*. cos. P C = sin. latitude X cos. A [1.] 

A C, being a measured distance on an arc of a great circle, must 
be reduced to the corresponding angle. 

Anrrlfl . • . , length of arc X 3437*7468 

Angle ot any arc in minutes = — 2 p . 

radius 




334 LAND-SUE V EYING. 

(3437-7468 = 57 '29598 X 60). Art. 280. 
Treating the earth as a sphere, this becomes : 

Angle of arc in minutes = length of arc -, ' ' , . 

c 2091240a 

Log. arc in minutes = log. length — 3*7941301 . . . [2.] 

Then use the value obtained by [2] in formula [1]. 

B C is found as an angle. To reduce it to feet, we have : 

an^le in minutes X radius 



Length in feet = 



3437-7468 



T „ . . angle in seconds X radius 
Length m feet = 60 X 3437-7468 " " 

Log. length in feet = log. angle in seconds + 2*0059789 . . [3.] 

485. Otherwise. Find the length of an arc subtending one 
second at the center, 

2ttX 20912405 im QO _ . , 
360~>^0ir60- = 101,386feet; 
i. e., 101 -386 feet subtends an angle of one second at the center of the 

,, ' mi , . , distance in feet ,. 

earth. Ihen, angle m seconds = , and distance = 

angle in seconds X 101*386 [4.] 

486. Approximately, 

B C in seconds = J P 2 (in seconds) X sin. 2P A X sin, 1". . [5.] 

tan. A B 



To find P. tan. P 



sin. AP* 



487. Example. Latitude 45° north, and distance 6 miles, re- 
quired the offset B C. 

6 miles = 31680 feet. 
By [2] log. 31680 = 4*5007852 

-3*7941301 

log. 5'*089265 = -7066551 

5'-089265 = 5' 5"'356 

By [1] log. sin. 45° = 9*8494850 

log. cos. 5' 5" *356 = 9*9999995 

log. cos. P C = log. cos. 45° 0' ,, *237 = 9*8494845 

.-. BC = A '*237 



THE SOLAR COMPASS. 335 

To reduce to feet by [3], log. 0"-237 = 1-3747483 

+ 2-005978 9 
log. B C in feet = log, 24-029 feet = 1-3807272 
Second Method : 

A ^ le = i™ = 312// = 5 ' 12 " 46a 
Then, as above, we find B C = // -237 of arc. 

B C in feet = 0"-237 X 101-386 = 24-0289 feet 
Approximate Method: 
Solving by formula [5], we find BC =24*3 feet. 



488. Spheroidal Formula. The preceding methods suppose the 
earth to be a sphere. Treating it as a spheroid, the following for- 
mula is without material error for distances within 100 miles : 

(1 - [e 2 . sin. 2 L])* 



CB = IF tan. L 



a 



h = distance in feet, L = latitude of initial point. 

a equatorial radius = 20926062 feet, 

e = -08169683. 

Example. Latitude 45° N. Distance 6 miles. 

log. e 2 = 3-8244104. 

(9-8494850 
log. sim 45 = | g . 8494850 



log. -0033718= 3-5233804 

1 — -0033718 = -9966283 

log. 0-9966283 = T'9992666 = log. numerator. 

log. -|- = 1-6989700 

4-5007852 



] o- Z* S 

s " ( 4-5007852 

log. tan. 45 = 10* 

log. numerator = 1-9992666 

8-6998070 

log. a = 7- 3206875 

log. 23-939 feet = 1-3791195 

489. Length of Parallels. The radius of any parallel of latitude 
equals the radius at the equator multiplied by the cos. latitude. 



336 LAND-SURVEYING. 

Then length in feet of 1° = —^. . radius in feet X cos. latitude. 

lbu 

Then length in feet of 1° = ~ X 20912405 X cos. latitude. 

log. length in feet of 1° =log. cos. latitude + 5-5622814. 
Example. To find the length of a degree on the 45° parallel. 
log. cos. 45 = 9-8494855. 
5-5622814 
log. 258087 = 5-4117669. 
Conversely. The angle, in minutes, subtended by any arc = 
length of arc X 3437*7468 

radius X cos. latitude 
log. angle in minutes = log. arc in feet — 3-7841301 — cos. latitude. 
Example. Latitude 45° N. and distance 6 miles, 
log. 31680 = 4-5007852 
- 3-7841301 



•7166551 

co-log. cos. 45° = -1505150 

log. 7 21" -897 -8671701 

490. The difference of lengths of any two parallels is called the 
convergence of the meridians between those parallels. This may 
be obtained more easily, since the distances between the meridians 
are as the cosines of the latitudes. 

Example. Two "range-lines" (meridians) are 6 miles (480 
chains) apart on the base-line of 46°. 

Required their convergence at 47° north. 

Pft q 47 
Length at 47° = 480 * ' = 471-252. 
° cos. 46 

480 - 471-252 = 8 chains 74*8 links. 



PAET II. 
LEVELING 



INTRODUCTION. 

491. Leveling in General. A level surface is one which is every- 
where perpendicular to the direction of gravity, as indicated by a 
plumb-line, etc., and consequently parallel to the surface of stand- 
ing water. It is, therefore, spherical (more precisely, spheroidal), 
but, for a small extent, may be considered as plane. Any line 
lying in it is a level line. 

A vertical line is one which coincides with the direction of 
gravity. 

The height of a point is its distance from a given level surface, 
measured perpendicularly to that surface, and therefore in a ver- 
tical line. 

Leveling is the art of determining the difference of the heights 
of two or more points. 

To obtain a level surface or line, usually the latter, is the first 
thing required in leveling. 

When this has been obtained, by any of the methods to be here- 
after described, the desired height of a point may be determined 
directly or indirectly. 

492. Direct Leveling. In this method of leveling, a level line 
is so directed and prolonged, either actually or visually, as to pass 
exactly over or under the point in question— i. e., so as to be in 
the same vertical plane with it— and the height (or depth) of the 
point above (or below) this level line is measured by a vertical rod, 
or by some similar means. The height of any other point being 



338 LEVELING. 

determined in the same manner, the difference of the two will be 
the height of one of the points above the other. So on, for any 
number of points. 

Dieect Leveling is the method most commonly employed. 
It will form Chapter I of this part. 

493. Indirect Leveling. In this method of leveling the desired 
height is obtained by calculation from certain co-ordinate meas- 
ured lines or angles, which fix the place of the point. 

Thus, the horizontal distance from any point to a tree being 
known, and also the angle with the horizon made by a straight line 
passing from the point to the top of the tree, its height above the 
point can be readily calculated. This is the most simple and most 
usual form of this method, though many others may be employed. 

Indikect Leveling will be developed in Chapter II. 

494. Barometric Leveling. This determines the difference of 
the heights of two points by the difference of the weights of the 
portions of the atmosphere which are above each of them, as indi- 
cated by a barometer. It is explained in Chapter III, 



CHAPTER I. 

DIRECT LEVELING. 
GENERAL PRINCIPLES. 

495. Leveling Instruments. The instruments employed to ob- 
tain a level line may be arranged in three classes, depending on 
these three principles : 

1. That a line perpendicular to a vertical line is a horizontal or 
level line. 

2. That the surface of a liquid in repose is horizontal. 

3. That a bubble of air, confined in a vessel otherwise full of a 
liquid, will rise to the highest point of that liquid. 

They will be described in the following pages. 

496. Methods of Operation. When a level line has been ob- 
tained, by any means, the difference of heights of any two points 
may be found by either of these two methods : 

First Method. Set the leveling instrument over one of the 

Fig. 346. 




■^m/TP?- 



points, as A, in Fig. 346. Measure the height of the level line 
above the point. Then direct this line to a rod held on the other 



340 



LEVELING. 




point, and note the reading. The difference of the two measure- 
ments at A and B will be the difference of their heights. 

Second Method. Let A and B, Fig. 347, represent the two 

points. Set the instru- 
ment on any spot from 
which both the points 
can be seen, and at such 
a height that the level 
line will pass above the 
highest one. Sight to a 
rod held at A, and note 
the reading. Then turn 
the instrument toward 
B, and note the height 
observed on the rod held at that point. The difference of the 
two readings will be the difference of the heights required. The 
absolute height of the level line itself is a matter of indifference. 

497. Curvature. The level line given by an instrument is tan- 
gent to the surface of the earth. Therefore, 
the line of true level is always below the line 
of apparent level. In Fig. 348, A D repre- 
sents the line of apparent level, and A B the 
line of true level. DB is the correction 
for the earth's curvature. By geometry we 
have : A D 2 = D B X (D B + 2 B 0). But 
D B, being very small, compared with the 
diameter of the earth, may be dropped from 
the quantity in the parenthesis, and we have : 

T)R- AD ° • 
DB -2BO ; 

i. e., the correction equals the square of the 
distance divided by the diameter of the earth. 
The difference of height for a distance of 
, ., 1 5-280 X 12 Q . , 

1 mile = 7916 = 7916 = 8 mclieS ' 
This varies as the square of the distance. The effect, if neg- 
lected, is to make distant objects appear lower than they really are. 



Fig. 348. 




PERPENDICULAR LEVELS. 



341 



7 



The effect is destroyed by setting the instrument midway be- 
tween the two points. 

498. Refraction. Rays of light coming through the air are 
curved downward. The effect is, to make objects look higher than 
they really are. Its amount is about one seventh that of curva- 
ture, and it operates in a contrary direction. 

PERPENDICULAR LEVELS. 

499. Principle. The principle upon which these are con- 
structed is, that a line perpendicular 
to the direction of gravity is a level 
line. 



Fig. 349. 




Fig. 350. 



fit 



500. Plumb-line Levels. The A 

level, Fig. 349, is so adjusted that, 

when the plumb-line coincides with 

the mark on the cross-piece, the feet of the level shall be at the 

same height. It is adjusted by reversion thus : Place its feet on 

any two points. Mark on the cross-bar 
the place of the plumb-line. Turn the 
instrument end for end, resting it on 
the same points, and mark the new place 
of the plumb-line. The point midway 

1 between the two is the right one. 

Another form is shown in Fig. 350. 
The above forms are not convenient for prolonging a level line. 

To do this, invert the preceding form, as in Fig. 351. 
To test and Fig 351 

adjust this, sight ^_ , 

to some distant 

point nearly on 

a level, and mark 

where the plumb- 
line comes to on 

the bottom of the 

rod. Turn the instrument around and sight again, and note the 

place of the plumb-line. The midway point is the right one. 



342 



LEVELING. 



A modification of the last form is to fasten a common carpen- 
ter's square in a slit in the top of a staff, by means of a screw, and 

then tie a plumb-line at the angle so that 
it may hang beside one arm. When it 
has been brought to do so, by turning 
the square, then the other arm will be 
level. 




501. Reflecting Levels. In these, the 
perpendicular to the direction of gravity 
is not an actual line, but an imaginary 
reflected line. 

It depends on the optical principle that a ray of light which 
meets a reflecting plane at right angles is reflected back in the 
same line. 

When the eye sees itself in a plane mirror, the imaginary line 
which passes from the eye to its image, is perpendicular to the mir- 
ror. Therefore, if the mirror be vertical, the line will be horizon- 
tal. It may therefore be used like any other line of sight for 
determining points at the same height as itself. 

The first form, Fig. 353 (Colonel Burel's), con- 
sists of a rhomb of lead, of about two inches on 
a side, and one inch thick. 

One side (the shaded part of the figure) is 
faced with a mirror. The right-hand corner of 
the rhomb is cut off, as seen in the figure, and a 
wire, A B, is stretched across the mirror. 

To use this, hold up the instrument, with the 
mirror opposite the eye, by the string D, so that 
the eye seems bisected in the mirror by the wire A B. Then 
glance through the opening at B, and any point in the line of the 
eye and wire will be in the same horizontal plane with them. 

The correctness of the instrument may be verified in the follow- 
ing manner : Hold up the instrument before any plane surface, as 
a wall, and determine the height of some point, as previously 
directed. Then, without changing the height of the instrument, 
turn it half around, place yourself between it and the wall, and 




PERPENDICULAR LEVELS. 



343 



Fig. 354. 



'Il«!!!ll 



note the point of the wall which is seen in the mirror to coincide 
with the image of the eye. 

If the two points on the wall coincide, the instrument is cor- 
rect. If they do not, the mirror does not hang plumb, and the 
point midway between the two is the true one. 

The instrument is rectified, or made to hang plumb, by means 
of the pear-shaped piece of lead seen attached to the lower corner 
of the rhomb. 

The second form consists of a hollow brass cylinder, with an 
opening at the upper end, as seen in Fig. 354. At the opening is 
a small mirror, whose ver- 
tical plane makes an an- 
gle with the vertical 
plane of section by which 
the cylinder was cut in 
forming the aperture. 
The edge of the mirror 
is marked thus (x) in the 
first half of Fig. 354. 
The mirror is made to 

hang plumb by means of a one-sided weight within the cylin- 
der. 

This is used by setting it on a stake driven into the ground, or 
by holding it in the hand, making the lower edge of the opening 
answer the same purpose as the wire in the other case. 

The same methods of verifica- 
tion and rectification are used as 
with the first form of the in- 
strument. 

The instrument, in its third 
form, is simply a small steel cylin- 
der, 4" or 5" long, and j-" in di- 
ameter, highly polished, and sus- 
\j) pended from the center of one end 

by a fine thread. 
To use this, hold it up by the thread with one hand, and with 
the other hand hold a card between the eye and instrument, using 




Fig. 355. 



1^5j) 



"CD 



g^ 



344 LEVELING. 

the upper edge of the card, as seen reflected in the mirror, the 
same as the wire in the first form. 

This instrument is the invention of M. Cousinery. 

WATER-LEVELS. 

502. Continuous Water-Levels. These may consist of a channel 
connecting the two points, and filled with water ; or of a tube, 
usually flexible, with the ends turned up, and extending from one 
point to the other. 

By measuring up or down, from the surface of the water at 
each end, the relative heights of the two points may be determined. 

503. Visual Water-Levels. The simplest one is a short surface 
of water prolonged by sights at equal distances above it, as in 
Fig. 356. 

A portable form is a tube bent up at each end, and nearly filled 

Fig. 356. 



with water. The surface of the water in one end will always be at 
the same height as that in the other, however the position of the 

tube mav vary. It may be 



Fig. 357. 



s&- 



I -^ 



easily constructed with a tube 
of tin, lead, copper, etc., by 
L) bending up, at right angles, 

f] an inch or two of each end, 

and supporting the tube, if 
too flexible, on a wooden bar. In these ends, cement (with putty, 
twine dipped in white-lead, etc.) thin vials, with their bottoms 
broken off, so as to leave a free communication between them. 
Fill the tube and the vials, nearly to their top, with colored 
water. Blue vitriol or cochineal may be used for coloring it. Cork 
their mouths, and fit the instrument, by a steady but flexible joint, 
to a tripod. 



AIR-BUBBLE OR SPIRIT LEVELS. 345 

To use it, set it in the desired spot, place the tube by eye 
nearly level, remove the corks, and the surfaces of the water in the 
two vials will come to the same level. Stand about a yard behind 
the nearest vial, and let one eye, the other being closed, glance 
along the right-hand side of one vial, and the left-hand ' side of 
the* other. Raise or lower the head till the two surfaces seem to 
coincide, and this line of sight, prolonged, will give the level line 
desired. Sights of equal height, floating on the water, and rising 
above the tops of the vials, would give a better-defined line. 

AIR-BUBBLE OR SPIRIT LEVELS. 

504. The "spirit-level" consists essentially of a curved glass 
tube nearly filled with alcohol, but with a bubble of air left within, 
which always seeks the highest spot in the tube, and will there- 
fore, by its movements, indicate any change in the position of the 
tube. Whenever the bubble, by raising or lowering one end, has 
been brought to stand between 

two marks on the tube, or, in 

case of expansion or contraction, ^"ZLZZZ 7 ■»■■"■■" - — -j-— 

to extend an equal distance on | 

i i 

either side of them, the bottom U 

of the block (if the tube be in 

one), or sights at each end of the tube, previously properly ad- 
justed, will be on the same level line. It may be placed on a board 
fixed to the top of a staff or tripod. 

When, instead of the sights, a telescope is made parallel to the 
level, and various contrivances to increase its delicacy and accu- 
racy are added, the instrument becomes the engineer's spirit-level. 

The upper surface of the tube is usually the arc of a circle, 
and, when we speak of lines parallel to a " level," we mean parallel 
to the tangent of this arc at its highest point, as indicated by the 
middle of the bubble. 

505. Sensibility. This is estimated by the distance which the 
bubble moves for any change of inclination. It is directly propor- 
tional to the radius of curvature of the tube. To determine the 
radius, proceed thus : 



346 



LEVELING. 



Let S == length of the arc oyer which the bubble moves for an 
inclination of 1 second (1"). 

Let R = its radius of curvature. 

Then S : 2ttR : : 1" : 360°, 
whence K = 206265 X S, 

orS - R 

• 206265 * 

S may be found by trial, the level being attached to a finely 

divided vertical circle. The 
FlG - S39 - radius may also be found 

without this, thus : Bring 
the bubble to center, and 
sight to a divided rod. 
Raise or lower one end of 
the level, and again sight to 
the rod. Call the difference 
of the readings 7i, the dis- 
tance of the rod d, and the 
space which the bubble 
Then we have two approximately similar triangles ; 

h ' 

Example. At 100 feet distance, the difference of readings was 
0*02 foot, and the bubble moved 0*01 foot. Then the radius was 
100 X 0-01 




moved S. 
whence r = 



0-02 



50 feet. 



The sensibility of an air-bubble level equals that of a plumb- 
line level having a plumb-line of the same length as the radius of 
curvature. 



506. Block-Level. If this is marked by the maker, and the bub- 
ble does not come to the center, when 
turned end for end, plane or grind 
off one end of the bottom until it 
does. 

Otherwise, if the bubble-tube is capable of movement, raise 
or lower one end of it until it will verify, bringing the bubble 



AIR-BUBBLE OR SPIRIT LEVELS. 



347 



half-way back to the middle by this means, and the other half 
by raising or lowering one end of the block, because the reversion 
has doubled the error. 

Eepeat this, if necessary. 



Fig. 361. 



Circular Level. The upper surface of this is spherical, 
therefore indicate a level in every direction, instead 
of only one, as does the preceding. It is adjusted 
like the last one, but in two directions, at right an- 
gles to each other. 



507. Level with Sights. The line of sight is made 
parallel to the tangent of the level. It may be tested thus 



It will 




m -. 


Fig. 362. 




4 - '"■ 


1 -- 


~~----< 


L 


■^=^" j 


L_ 


n ii . 



Bring the bubble to the center of the tube and make a mark, in 
the line of sight, as far off as can. be seen. Then turn the level end 
for end, and sight again. If the bubble remains in the same place, 
"all right." If not, rectify it by altering the sights, or by altering 
the marks for the bubble to come to, bringing the bubble half-way 
back, and trying it again. 



six inches long, and one inch in diameter. 



Fig. 363. 



508. Hand-Reflected Level. This consists of a brass tube, about 

To the inside of the 

upper portion of the 

tube is attached a 

small level. A small 

mirror is placed at an 

angle in the lower side 

of the tube, so that it will reflect the point to which the bubble 

must come, in order to have the instrument level, to the eye. 
23 




348 



LEVELING. 



A small hole at one end, and a horizontal cross-hair at the other, 
give the desired level line. It is used by holding it in the hand. 
Fig. 363 is an approved form, made by Young, of Philadelphia. 
The improvement consists in the patent " Locke sight," which 
enables the near cross-hair to be distinctly seen at the same time as 
the distant object. 

509. Gurley's Telescopic Hand-Level (Fig. 363', a). "This 
consists of a tube to which are fitted the lenses of a single opera- 



Fig. 363'. 





glass, and containing in addition thereto a reflecting prism, cross- 
wire, and small spirit-level, the last being shown in the open part 
of the tube. 

" The eye-lens, as indicated in the cut, is made of two separate 
pieces, the larger one being the usual concave eye-lens of the 
opera-glass, the smaller one a segment of a plano-convex lens hav- 
ing its focus in a cross-wire under the level-vial and above the 
reflecting prism. 

"The observer holds the tube horizontal, with the level open- 
ing uppermost, and with the same eye sees the object toward 
which the instrument is directed, and observes the position of the 
bubble. When the level is truly horizontal, the cross-wire will 



AIR-BUBBLE OR SPIRIT LEVELS. 349 

bisect the bubble, and will also determine the level of any object 
seen through the telescope. 

"In the binocular form of this level (Fig. 363', b) the tube 
on the right incloses the usual lenses of the opera-glass, while that 
on the left contains only the prism, level-vial, and cross-wire. 
The binocular hand-level gives a clearer view of an object than is 
possible with a single tube, there being no light lost by the inter- 
ference of the prism and level-vial." 

510. The Telescope-Level. In this the line of collimation of 
the telescope corresponds to the sights of Fig. 362, and is made 
parallel to the level — i. e., this line is so adjusted as to be horizon- 
tal when the bubble of its level is in the center. 

There are many different forms of the telescope-level, of which 
the most important ones will now be given. 

511. The Y-Level. This is so named from the shape of the 
supports of the telescope. It is the variety most used by American 
engineers. 

Fig. 364 represents a Y-level of the usual form. The telescope 
is held in the wyes by the clips, A A, which are fastened to the 



Fig. 364. 




wyes by tapering pins, so that the telescope can be clamped in any 
position. The milled-headed screws at M and M are used to move 



350 



LEYELIXG. 



Fig. 365. 



the object-glass and eye-piece in and out, so as to adjust them for 
long and short sights, and for short-sighted and long-sighted per- 
sons. L is a spirit-level ; P and P are parallel 
plates ; C is the clamp-screw, which fastens 
the spindle on which the level-bar, B, which 
supports the wyes, turns ; T is the tangent- 
screw, by which the telescope may be slowly 
turned around horizontally. 

512. The Telescope. The arrangement of 
the parts of the telescope is shown in Fig. 365. 
is the object-glass, by which an image of 
any object, toward which the telescope may be 
directed, is formed within the tube. E E is 
the eye-piece — a combination of lenses, so ar- 
ranged as to magnify the small image formed 
by the object-glass. The cross-hairs are at X. 
They are moved by means of the screws shown 

p, at B B. A A are screws used for centering the 
eye-piece. C C are screws used for centering 
the object-glass. At D D are rings, or collars, 
of exactly the same diameter, turned very tru- 
ly, by which the telescope revolves in the wyes. 
The telescope shown in the figure forms the 
image erect. Other combinations of lenses are 
used, some of which invert the image ; but the 
one here shown is generally preferred. 

513. The Cross-Hairs. These are made of 
very fine platinum wire or of spider-threads. 
They are attached to a short, thick tube, 
placed within the telescope-tube, through which 
pass loosely four screws whose threads enter 
and take hold of the cross-hair ring, as shown 
in Fig. 366. 

In some instruments, one of each pair of 
opposite screws is replaced by a spring ; and the screws, instead 



w> 



u 



/"I 



:u 



AIR-BUBBLE OR SPIRIT LEVELS. 351 

of being capstan -headed, and moved by an " adjusting-pin," have 

square heads, and are moved by a " key," like a watch-key. 

The line of collimation 

Fig. 366. 




(or line of aim) is the 
imaginary line passing 
through the intersection of 
the cross-hairs and the op- 
tical center of the object- 
glass. 

The image formed by 
the object-glass should co- 
incide precisely with the 
cross-hairs. When this is 
not the case, there will be an apparent movement of the cross- 
hairs, about the objects sighted to, on moving the eye of the ob- 
server. This is called instrumental parallax. To correct it, 
move the eye-piece out or in, till the cross-hairs are sharply de- 
fined against any white object. Then move the object-glass in 
or out, till the object is also distinctly seen. The image is now 
formed where the cross-hairs are, and no movement of the eye will 
cause any apparent motion of the cross-hairs. 

514. The Level. This consists of a. thick glass tube, slightly 
curved upward, and so nearly filled with alcohol that only a small 
bubble of air remains in the tube. This always rises to the highest 
part. The brass case, in which this is inclosed, is attached to the 
under side of the telescope, and is furnished with the means of 
moving, at one end vertically, and at the other horizontally. Over 
the aperture, in the case, through which the bubble-vial is seen, 
is a graduated level-scale, numbered each way from zero at the 
center. 



515. Supports. The wyes in which the telescope rests are sup- 
ported by the level-bar, B, and fastened to it by two nuts at each 
end (one above, one below the bar), which may be moved with an 
adjusting-pin. The use of these nuts will be explained under "Ad- 
justments." Attached to the center of the level-bar is a steel 



352 



LEVELING. 



spindle, made so as to turn smoothly and firmly in a hollow cylin- 
der of bell-mefcal ; this, again, is fitted to the main socket of the 
upper parallel plate. 

516. Parallel Plates. It is by the aid of these that the instru- 
ment is leveled. The plates are united by a ball-and-socket joint, 




and are held apart by the four plate-screws, QQQQ. which pass 
through the upper one, and press against the lower one. 



AIR-BUBBLE OR SPIRIT LEVELS. 



353 



To level the instrument, turn the telescope till it is brought 
over a pair of opposite parallel plate-screws. Then turn the pair 
of screws, to which the telescope has been made parallel, equally 
in opposite directions, 
screwing one in and the 
other out, till the bub- 
ble is brought to the 
center. Then turn the 
telescope so as to bring 
it over the other pair of 
opposite screws, and 
bring the bubble to the 
center, as before. 

Repeat the opera- 
tion, as moving one pair 
of screws may affect the 
other. 

Sometimes one of ° 

a 

each pair of opposite c 

pi 
screws is replaced by a 

strong spring, and in 
some instruments only 
three screws are used. 

The lower plate is 
screwed on to the tripod- 
head. 

517. Fig. 367 is a 
twenty - inch Y - level, 
and Fig. 368 is a longi- 
tudinal section of it, 
showing its construc- 
tion. 

In Fig. 368, B B are 
the screws attached to the cross-hair ring. At A are four screws 
holding a ring through which the inner end of the eye-piece passes. 
At C are four screws holding a ring, through which the inner 




354 LEVELING. 

end of the object-glass slide passes. The use of these sets of screws 
will be explained under "Adjustments." 

The interior spindle, D, which supports the instrument, and on 
which it turns, is made of steel, and is carefully fitted to the in- 
terior of a hollow socket of bell-metal, which has its exterior sur- 
face fitted to the main socket, E, of the tripod-head. The hollow 
bell-metal socket is held in place by a washer and screw, shown 
at D. 

A screw, passing through the main socket, E, enters a groove 
in the exterior of the bell-metal socket, and fastens the instrument 
to the tripod-head. 

ADJUSTMENTS. 

518. The line of collimation of the telescope should be horizon- 
tal when the bubble is in the center of the tube ; which will be the 
case when this line is parallel to the plane of the level. But both 
this line and this plane are imaginary, and can not be compared 
together directly. They are therefore compared indirectly. The 
line of collimation is made parallel to the bottom of the collars, 
and the plane of the level is then made parallel to them. 

519. First Adjustment. To make the line of collimation parallel 
to the bottoms of the collars. 

Sight to some well-defined point, as far off as it can be dis- 



tinctly seen. Then revolve the telescope half around in its sup- 
ports — i. e., turn it upside down. If the line of collimation was 
not in the imaginary axis of the rings, or collars, on which the 
telescope rests, it will now no longer bisect the object sighted to. 
Thus, if the horizontal hair was too high, as in Fig. 369. this line 
of collimation would point at first to A, and, after being turned 
over, it would point to B. The error is doubled by the reversion, 
and it should point to C, midway between A and B. Make it do 



ADJUSTMENTS. 355 

so, by unscrewing the upper capstan-headed screw, and screwing in 
the lower one, till the horizontal hair is brought half-way back to 
the point B. Bemember that, in an erecting telescope, the cross- 
hairs are reversed, and vice versa. Bring it the rest of the way by 
means of the parallel plate-screws. Then revolve it in the wyes 
back to its original position, and see if the intersection of the cross- 
hairs now bisects the point, as it should. If not, again revolve, 
and repeat the operation till it is perfected. If the vertical hair 
passes to the right or to the left of the point when the telescope is 
turned half around, it must be adjusted in the same manner by the 
other pair of cross-hair screws. One of these adjustments may 
disturb the other, and they should be repeated alternately. When 
they are perfected, the intersection of the cross-hairs, when once 
fixed on a point, will not move from it when the telescope is re- 
volved in its supports. This double operation is called adjusting 
the line of collimation. 

It has now been brought into the center line, or axis, of the 
collars, and is therefore parallel to their bottoms, or the points on 
which they rest, if they are of equal diameters. We have to assume 
this as having been effected by the maker. 

In making this adjustment, the level should be clamped, but 
need not be leveled. 

520. Second Adjustment. To make the bottoms of the collars 
parallel to the plane of the level — i. e. , to insure their being hori- 
zontal when the bubble is in the center. 

Clamp the instrument, and bring the bubble to the center by 
the parallel plate-screws. Take the telescope out of the wyes, and 
turn it end for end. If the bubble returns to the center, " all 
right." If not, rectify it, by bringing the bubble half-way back, 
by means of the nuts which are above and below one end of the 
bubble-tube, and which work on a screw. Bring it the rest of the 
way by the plate-screws, and again turn end for end. Repeat the 
operation, if necessary. 

If, in revolving the telescope (as in the first adjustment), the 
bubble runs toward either end, it must be adjusted side wise, by 
means of two screws which press horizontally against the other end 



856 LEVELING. 

of the bubble-tube. This part of the adjustment may derange the 
preceding part, which must, therefore, be tried again. 

521. Third Adjustment. To cause the bubble to remain in the 
center of the tube when the telescope is turned around horizontally. 

To verify this, bring the bubble to the center of the tube, and 
then turn the telescope half-way around horizontally. If the bub- 
ble does not remain in the center, adjust it by bringing it half-way 
back by means of the nuts at the end of the level-bar. Test it by 
bringing it the rest of the way back by the parallel plate-screws, 
and again turning half-way around. 

The cause of the difficulty is, that the plane of the level is not 
perpendicular to the axis about which it turns, and that this axis 
is not vertical. The above operations correct both these faults. 

This adjustment is mainly for convenience, and not for accu- 
racy, except in a very small degree. 

Some instruments have no means of making the third adjust- 
ment. They must be treated thus : 

Use the screws at the end of the bubble-tube, to cause the bub- 
ble to remain in the center when the level is turned around hori- 
zontally. Then make the line of collimation parallel to the level 
by raising or lowering the cross-hairs. 

522. When levels are provided with the means of centering the 
eye-piece and object-glass, these operations should precede the first 
three which we have just explained. 

• Centering the Object- Glass. After adjusting the line of collima- 
tion for a distant object (as explained in the " First Adjustment 7 ') 
move out the slide, which carries the object-glass, until a point ten 
or fifteen feet distant can be distinctly seen. Then turn the tele- 
scope half over, as before, and see if the intersection of the cross- 
hairs bisects the point. If not, bring it half-way back by the 
screws C 0, Fig. 365, moving only one pair of screws at a time. 
Repeat the operation for a distant point, and then again for a near 
one, if necessary. We have now adjusted the line of collimation 
for long and short sights, and may assume it to be in adjustment 
for intermediate ones, since the bearings of the slides are supposed 
to be true, and their planes parallel to each other. 



ADJUSTMENTS. 



357 



Centering the Eye- Piece. This is to enable the observer to see 
the intersection of the cross-hairs precisely in the center of the*'field 
of view of the eye-piece. It is adjusted by means of four screws, 
two of which are shown at A A. 

These operations are performed by the maker so permanently 
as to need no further attention from the engineer, and the heads of 
the screws, by which these adjustments are made, are covered by a 
thin ring which protects them from disturbance. 

523. Adjustment by setting between two points, or the " Peg- 
Method" Drive two pegs several hundred feet apart, and set the 
instrument midway between them. Level, and sight to the rod 
held on each peg. The difference of the readings will be the true 
difference of the heights of the pegs, no matter how much the level 
may be out of adjustment. 

Then set the level over one peg, and sight to the rod at the 
other. Measure the height of the cross-hairs above the first peg. 
The difference of this and the reading on the rod should equal the 
difference of the heights of the two points, as previously deter- 
mined. If it does not, set the target to the sum or difference of 
the height of the cross-hairs above the first peg, and the true dif- 
ference of height of the points, according as the first point is higher 
or lower than the second, and hold the rod on the second point. 
Sight to it, and raise or lower one end of the bubble-tube until the 
horizontal cross-hair does bisect the target when the bubble is in 
the center. Then perform the " third adjustment." 

Instead of setting over one peg, it is generally more convenient 



Fig. 370. 




2.994 
2.398 



to set near to it, and sight to a rod held on it, and use this reading 
instead of the measured height of the cross-hairs. 



358 



LEVELING. 



N. B. —This verification should always be used for every level, 
even after the three usual adjustments have been made ; for it is 
independent of the equality of the collars. 

In running a long line of levels, let the last sight at night be 
taken midway between the last two " turning-point " pegs, and in 
the morning try their difference by setting close to the last one. 
This tests the level every day with very little extra labor. 

524. Egault's Level. In this level the bubble-tube is not con- 
nected with the telescope. It is used thus : 

Level and sight as usual. Then turn the telescope upside down, 
end for end, and half-way around horizontally, and sight again. 
Half the sum of the two readings is the correct one, no matter how 
much the instrument is out of adjustment (assuming the collars to 

be of equal size) ; for 
the errors then cancel 
each other. This is the 
one used principally in 

1 1 • The rod used with it 

is marked with numbers 
only half the real heights 
above its bottom. Then 

the sum of the readings is the true one. Thus the rod itself 

takes the mean of the readings. 



Fig. 371. 





Fig. 372. 




525. Troughton's Level. In this the bubble -tube is perma- 
nently fastened in the top of the telescope-tube. It is adjusted by 
the " peg method," or some similar one, the cross-hair being moved 
up or down until the observation gives the true difference of height 
of the pegs when the bubble is in the center. Then make the 
" third adjustment,'' by means of the screws under the telescope. 



ADJUSTMENTS. 



359 



Fig. 373. 




526. Gravatt's Level, or the "Dumpy Level." Its diameter is 
very great, thus giving more light. Its bubble is on the top, and 
can be seen in a small in- 
clined mirror, by the ob- 
server. It also has a 
cross-level. 

527. Lenoir's Level. In 

this, the telescope carries, 
at each end, a steel block, 
whose upper and lower 
faces are made perfectly 
parallel. They are placed 
on a brass circle, which 
is made level by reversing a level placed upon the upper sur- 
face of the steel blocks. 

528. Tripods. These consist of three legs, shod with iron, and 
connected by joints at the top. There are many different forms, 

the most common of 

Fig. 374. . . 

which is given m 

Fig. 367. Other 
forms are given in 
Art. 476. Lightness 
and stiffness are the 
desired qualities. 

Stephenson's tri- 
pod has a ball-and- 
socket joint below 
the parallel plates, so as to admit of being at once set nearly level 
on very steep slopes. 

" Quick-leveling " tripod-heads, for quickly setting the leveling- 
plates nearly level, are made of various patterns. 

Extension tripods are manufactured which provide for length- 
ening and shortening the legs of the tripod. 

529. Rods. These should be made of light, well-seasoned wood. 
A plumb or level attached to them will show when they are held 




360 



LEVELING. 



vertically. To detect whether the rod leans to or from the instru- 
ment, its front may be angular or curved. If angular, when held 
leaning toward the instrument, the lines of di- 
vision will appear as in Fig. 375. When lean- 
ing from the instrument, they will appear as in 
Fig. 376. They are usually divided to feet, 
tenths, and hundredths. 



Fig. 375. Fig. 376 
I 



530. Target. This is a plate of iron or 
brass, attached to the rod in such a way that it 
may be moved up and down the rod and 
clamped in any position. The face of the tar- 
get should be painted of such a pattern that, when sighting to 
it, it maybe very precisely bisected by the horizontal cross-hair. 
Some of the many varieties are given in Figs. 377-385. 

Those represented in Figs. 377, 378, and 379 are bad, because 



Fig. 377. 




Fig. 378. 



Fig. 379. 



<-W & 



Fig. 380. 




Fig. 3S1. 




Fig. 38: 



® 



Fig. 383. 




Fig. 381. 



Fig. 385. 




the cross-hair may be above or below the middle of the target by 
its full thickness, as magnified by the eye-piece of the telescope 
without the error being perceptible. The next three, Figs. 3S0. 



ADJUSTMENTS. 



361 



Fig. 38*7. 



Fig. 388. 



381, and 382, depend upon the nicety with which the eye can de- 
termine if a line bisects an angle. Fig. 383 depends upon the 
accuracy with which the 
eye can bisect a space. flMUJll) 

Fig. 384 depends upon, 
the accuracy with which 
the eye can bisect a 
circle. Figs. 381, 382, 
and 385 are the best 
forms for use. Red and 
white are the best col- 
ors. 



531. Vernier. The 

target carries a vernier, 
by which smaller spaces 
may be measured than 
those into which the 
rod is divided. It may 
be placed on the side of 
an aperture, in the face 
of the target, through 
which the divisions on 
the rod can be seen, or 
carried on the back or 
side of the rod by the 
target-clamp. 

532. The New York 

Rod (Fig. 386). This 
is usually in two pieces, 
sliding one upon the 
other, and connected by 
a tongue. It is gradu- 
ated to tenths and hun- 
dredths of a foot, and 
can be read to thousandths by the vernier 




Up to six feet and a 



362 



LEVELING. 



half the target is used as on other rods. For greater heights, the 
target is fixed at six and a half feet, and the back part of the rod, 
which carries the target, is shoved up (Fig. 386) until the target is 
bisected by the cross-hairs. Its height is then read off on the 
side of the rod, on which the numbers run downward, and on 
which is a second vernier, which gives the precise 
reading. It is convenient for its portability, but apt 
to be too tight or too loose, as the weather is moist 
or dry. Sometimes it is in three pieces, as in Fig. 
387. 



Pig. 




n 





533. The Boston Rod (Fig. 388). This is usually 
in two parts, like the Xew York rod. The target is 
rectangular, and is fastened to one of the pieces near 
its extremity. For heights less than six feet, the rod 
is held with the target-end down, and the target is 
moved up by sliding up the piece which carries it. 
For heights above six feet, the rod is turned end for 
end, bringing the target-end up, and then sliding up 
the piece which carries the target. 

534. The Philadelphia Rod (Fig. 389). This is 
in two parts, held together by brass clamps, and is 
furnished with a target. It is graduated and painted 
so as to be used as a " speaking-rod,*' or with a tar- 
get. TThen the target is used, the vernier on the 
target is read for height up to seven feet. For 
greater heights, the target is clamped at seven feet, 
and the part to which the target is clamped is slid 
up, and the vernier on the upper clamp is used. 

535. Speaking-Rods. These are rods which are 
read without targets, the divisions and subdivisions 
being painted on the face of the rod. They produce 
great saving of time and increase of accuracy. 

In one form (Fig. 390) the face of the rod is di- 
vided into tenths of feet, and smaller divisions estimated. 
In Bourdaloue's rod the divisions are each four centimetres 



ADJUSTMENTS. 



363 



Fig. 390. 



(1*6 inch), and are numbered at 
them as in Fig. 391. 
Gravatfs Rod 
(Fig. 392). This is 
divided to O'Ol foot. 
The upper hundredth 
of each tenth extends 
across the rod. Each 
half-tenth is marked 
by a dot ; each half- 
foot by two dots. 
Every other tenth is 
numbered, and the 
numbers are each 0*1 
high. It is in three 
parts, which slide into 
each other like a tele- 
scope. 

Barlow's Rod (Fig. 
393). In this the di- 
visions are marked by triangles, 

Fig. 393. 



half their value. He arranges 



Fig. 391. 



~ 



Fig. 


392. 




■win 


See 


- 




* 


= 






< 


= 


- 


P= 


■ 



24 





Fig. 394. 

— , — r- 






^ 




!k 














"2 


\ 








J^ 










^ 


S 


~---. 






^ 




^-^ 




?. 


-~-~~ 


\ 


-^■^ 


-—" 




""' 




>-— 


"--— . 


1 


4- 


■ — 


M 


"1 



each 0*02 foot high, so that it 
reads to hundredths, and less 
by estimation. This is based 
on the power the eye has in bi- 
secting angles. 

Stephenson's Rod (Fig. 394). 
This is based upon the princi- 
ple of the diagonal scale. Each 
tenth is bisected by a horizon- 
tal line, and the diagonals en- 
able the observer to read to 
hundredths. 

Conyb'eare's Rod (Fig. 395). 
It reads to hundredths of a 
foot by means of the cross-hair 
bisecting the tops and bottoms 
and angles of hexagons. The 



364 



LEVELING. 



Fig. 395. 



Fig. 396. 



odd tenths are made white and the even ones black. The figures 
are placed so that their centers are opposite the divisions they 

refer to. 

Pembertorts Rod (Fig. 396). 
This is on the principle of nine 
verniers placed side by side. 
It reads to hundredths, which 
are given by counting up from 
the dot which the hair bisects, 
to the dot in the same vertical 
line which is bisected by one of 
the horizontal lines which mark 
the tenths. The inventor 
claims that it can be read nine 
times as far as Gravatt's. 

On all speaking - rods, to 
avoid confounding numbers, such as 3 and 8, they may be marked 
thus : 

1 . 2 . Ill . 4 . V . 6 . 7 . 8 . IX . X . 11 . XII. 
The French, who go by tenths, use the following : 
1 . 2 . T . 4 . V , 6 . 7 . 8 . X . X. 
The figures are sometimes placed with their tops on a level with 
the tops of the dimensions they mark— e. g., feet ; and sometimes 
with their middles on the dividing line. 




i 

A* t 



THE PRACTICE. 
536. Field Routine ; or, how to start and go on : 

1. The rodman holds the rod on the starting-point, which may 
be a peg, a door-sill, or other "bench-mark." He stands square 
behind his rod, and holds it as nearly vertical as possible. 

2. The leveler sets up the instrument, somewhere in the direc- 
tion in which he is going, but not necessarily, or usually, in the 
precise line. He then levels the instrument by the parallel plate- 
screws, sights to the rod, and notes the reading, whether of target 
or speaking-rod, as a " back-sight " (B. S.), or + (plus) sight; 
entering it in the proper column of one of the tabular forms of 
field-book, given in the following articles. 



THE PRACTICE. 365 

3. The rodman is then sent ahead about as far as he was be- 
hind, and he there drives a "level-peg" nearly to the surface of 
the ground, or finds a hard, well-defined point, and holds the rod 
upon it. 

4. The leveler then again sights to the rod, and notes the read- 
ing as a "fore-sight" (P. S.), or — (minus) sight. The difference 
of the two readings is the difference of the heights of the points. 

5. He then takes up the instrument, goes beyond the rod, any 
convenient distance, sets up again, and proceeds as in paragraph 2 ; 
and so on for any number of points, which will form a series of 
pairs. The successive observations of each pair give their differ- 
ence of heights, and the combination of all these gives the differ- 
ence of heights of the first and last points of the series. 

6. If the vertical cross-hair be strictly vertical, it will determine 
whether the rod leans to the right or left. To know whether the 
cross-hair is vertical or not, try whether it coincides with a plumb- 
line, or sight to some fixed point, turn the telescope from side 
to side horizontally, and see if the horizontal cross-hair continues 
to cover the spot. If it does not, turn the telescope around 
in the wyes till it does ; then it is truly horizontal, and the other 
hair, being perpendicular to it, is truly vertical. To know whether 
the rod leans forward or backward, have the rodman move it from 
and to himself. If the line bisected by the cross-hair descends in 
both motions, the rod was vertical ; if the line rises, the rod was 
leaning. The lowest reading is the true one. 

7. When a target is used, signals are made by the leveler with 
the hand, "up "and " down," to indicate in which direction to 
move the target. Drawing the hand to the side signifies " stop," 
and both hands brought together above the head signifies " all 
right." The rodman should move the target fast at first, and 
slowly after having passed the right point. When signaled " all 
right," he should clamp the target and show again. Then call out 
the reading before moving, and show it to the leveler, as either 
passes the other. 

8. We have thus far supposed that only the difference of heights 
of the two extreme points is desired. But when a section or profile 
of the ground is required, the rod must be held and observed, at 



366 LEVELING. 

each change of slope of the ground, or at regular distances ; usu- 
ally, for railroad-work, at every hundred feet, and also at any 
change of slope between those points. 

Any number of points, within sight, may have their relative 
heights determined at one setting of the level. 

The names back-sight (B. S.) and fore-sight (F. S.) do not 
necessarily mean sights taken looking forward or backward (though 
they are generally so for turning-points), but the first sight taken, 
after setting up the instrument, is a B. S. or -f- (plus) sight, and all 
following ones, taken before removing the instrument, are F. S.'s, 
or — (minus) sights. The full meaning of this will appear in con- 
sidering the forms of field-book. 

All but the first and last points sighted to are called interme- 
diate points, or "intermediates." The last point sighted to before 
moving the instrument is called a turning-point, or clianging- 
point. 

The first and last sights, taken at any one setting of the instru- 
ment, require the greatest possible accuracy. The intermediate 
points may be taken only to the nearest tenth, or hundredth at 
most ; because any error in them will not affect the final result, 
but only the height of that single point at which it was taken. 

Two rodmen are often used to save the time of the leveler. 
Then it is well to use a target-rod for the " turning-points," which 
are often distant and need most precision, and a speaking- rod for 
the intermediate points. Where one rod is used, the rodman 
should keep notes of the readings at the turning-points. 

537. Field-Notes. The beginner may sketch the heights and 
distances measured, in a profile or side view, as in Fig. 397. But 
when the observations are numerous, they should be placed in one 
of the tabular forms given on the following pages. 

538. First Form of Field-Book. In this, the names of the points 
or " stations," whose heights are demanded, are placed in the first 
column, and their heights, as finally ascertained, in reference to 
the first point, in the last column. The heights above the starting- 
point are marked -f-, and those below it are marked — . The back- 



THE PRACTICE. 



367 



sight to any station is placed on the line below the point to which 
it refers. When a back-sight exceeds a fore-sight, their difference 

Fig. 391 




is placed in the column of " Eise " ; when it is less, their difference 
is a " Fall." The following table represents the same observations 
as the last figure, and their careful comparison will explain any 
obscurities in either : 



STATIONS. 


DISTANCES. 


BACK- 
SIGHTS. 


FORE- 
SIGHTS. 


RISE. 


FALL. 


TOTAL 
HEIGHTS. 


A 

B 
C 
D 
E 
F 


100 
60 
40 
70 
50 


2-00 
3-00 
2'00 
6-00 
2-00 


6'00 

4-00 
1-00 
1-00 
6*00 


+ 1-00 
+ 5-00 


—4-00 
— 1-00 

—4-00 


O'OO 

—4-00 
—5-00 
—4-00 
+ 1-00 
—3-00 


15-00 


18-00 


—3-00 



The above table shows that B is 4 feet below A ; that C is 5 
feet below A ; that E is 1 foot above A ; and so on. To test the 
calculations, add up the back-sights and fore -sights. The differ- 
ence of the sums should equal the last "total height." 

An objection to this form is that the back-sights come on the 
line Mow the station to which they are taken, which is embarrass- 
ing to a beginner. 

When " intermediate " observations are taken, the " fore- 
sights " taken to these intermediate points are put down in their 
proper column, and are also set down in the column of "back- 
sights " ; so that, when the two columns are added up, any error in 



368 



LEVELING. 



these intermediate sights (which are usually not taken very accu- 
rately) will be canceled, and will not affect the final result. The 
effect is the same as if, after the fore-sight to the intermediate point 
had been taken, the instrument had been taken up and set down 
again at precisely the same height as before, and a back-sight had 
then been taken to the same point. Hence, in this form, the 
•' turning-points " are those stations which have different back- 
sights and fore-sights, while those which have them the same are 
"intermediates.'' 

The following figure and table represent the same ground as the 



Fig. 398, 




preceding one, but with only two settings of the instrument. D is 
the turning-point : 



STATIONS. 


DISTANCES. 


BACK- 
SIGHTS. + 


FORE- 
SIGHTS. — 


RISE. 


FALL. 


TOTAL 
HEIGHTS. 


A 
B 
C 
D 

E 
F 




2-00 
6-00 
7-00 
9-00 
4-00 


6-00 
7-00 
6-00 
4-00 
8-00 


1-00 
5-00 


4-00 
1-00 

4-00 


o-oo 

-4-00 
—5-00 
-4-G0 
+ 1-00 
-3-U0 


+ 28-00 


—31-00 


3 00 



In leveling for "sections," the distances between the -points 
leveled must be recorded. They are usually put down after the 
stations to which they are measured ; although in surveying with 
the compass, etc., they are put down after the stations from which 
they are measured. In the following notes, which contain inter- 



THE PRACTICE. 



369 



mediate stations, they are put down before the stations to which 
they are measured. It should be remembered that these distances 
are measured between the points at which the rod is held, and have 
no reference to the points at which the instrument is set up : 



DISTANCES. 


STATIONS. 


BACK- 
SIGHTS. + 


FORE- 
SIGHTS. — 


RISE. 


FALL. 


TOTAL 
HEIGHTS. 




260 








91-397 


100 


261 


4-576 


3-726 


0-850 




92-247 


100 


262 


5-420 


4-500 


0-920 




93-167 


100 


263 


4-500 


3-170 


1-330 




94-497 


40 


263-40 


4-910 


4-938 




0-028 


94-469 


60 


264 


4-938 


6-386 




1-448 


93-021 


100 


265 


3-380 


4-640 




1-260 


91-761 


100 


266 


4-640 


5-400 




0-760 


91-001 


70 


266-70 


2-760 


3-070 




0-310 


90-691 


30 


267 


3-070 


3-750 




0-680 


90-011 


100 


268 


6-750 


5-925 




3*175 


86-836 


41-944 


46-505 


-4-561 








41-944 




+ 91-397 




—4-561 


86-836 



539. Second Form of Field-Book. This is presented below. It 
refers to the same stations and levels noted in the first table, and 
shown in Fig. 397 : 



STATIONS. 


DISTANCES. 


BACK- 
SIGHTS. 


HEIGHT OF 

INSTRUMENT 

ABOVE DATUM. 


FORE- 
SIGHTS. 


TOTAL 
HEIGHTS. 


A 
B 

n 

<j 

D 
E 
F 


100 

60 
40 

70 
50 


2-00 
3-00 
2-00 
6-00 
2-00 


+ 2-00 

— 1-00 

— 3-00 
+ 2-00 
+ 3-00 


6-00 
4-00 
1-00 
1-00 
6-00 


o-oo 

—4-00 
—5-00 
—4-00 
+ 1-00 
—3-00 


15-00 


18-00 


—3-00 



In the preceding form it will be seen that a new column is in- 
troduced, containing the height of the instrument — i. e., of its line 
of sight — not above the ground where it stands, but above the 
Datum, or starting-point, of the levels. The former columns of 
"rise "and "fall "are omitted. The preceding notes are taken 
thus : The height of the starting-point, or " datum," at A, is 0*00. 
The instrument being set up and leveled, the rod is held at A» 



370 



LEVELING. 



The back-sight upon it is 2'00 ; therefore the height of the instru- 
ment is also 2 '00. The rod is next held at B. The fore-sight to 
it is 6*00. That point is therefore 6*00 below the instrument, or 
2*00 — 6*00 = — 4 # 00 below the datum. The instrument is now 
moved, and again set up, and the back-sight to B, being 3*00, the 
height of the instrument is — 4-00 + 3*00 = — 1*00, and so on ; 
the height of the instrument being always obtained by adding the 
back-sight to the height of the peg on which the rod is held, and 
the height of the next peg being obtained by subtracting the fore- 
sight to the rod held on that peg, from the height of the instrument. 

This form is better than the first form, in leveling for a section 
of the ground to make a profile ; or when several observations are 
to be made at one setting of the level ; or when points of desired 
heights are to be established, as in " leveling-location." 

This form may be modified by putting the back-sights on the 
same line with the stations to which they are taken. This avoids 
the defect of the first form, but introduces the new defect of writ- 
ing them down after the number which they precede, in a back- 
handed way, which may be a source of error. 

This modification is shown in the following table, which cor- 
responds to Fig. 398. In the column of fore-sights, the "turn- 
ing-points" (T. P.), and "intermediate points" (Int.), are put in 
separate columns ; so that, to prove the work, the difference of the 
sum of the back-sights and of the sum of the turning-point fore- 
sights, is the number which should equal the difference of the 
heights of the first and last points : 



STATIONS. 


DISTANCES. 


BACK- 
SIGHTS. + 


HEIGHT OF 


FORE-SIGHTS. — 


TOTAL 
HEIGHTS. 




T. P. 


INT. 


A 
B 
C 
D 
E 
F 




2-00 
9-00 


+ 2-00 
+ 5-00 


6-00 
8-00 


6-00 
7-00 

4-00 


o-oo 

-4-00 
—5-00 
—4-00 
+ 1-00 
-3-00 


+ 11-00 


—14-00 

+ 11-00 


— 300 



THE PRACTICE. 



371 



When a line is divided up into stations of 100 feet each, as on 
railroad- work, the number of the station indicates its distance from 
the starting-point. When an observation is taken at a point be- 
tween these hundred-feet stations, it is noted as a decimal, thus : 
Station 4*60 is 460 feet from the starting point. In the field-notes 
of such work, the column of distances may be omitted, as in the 
following table. The heights and distances are the same as in the 
last table under Art. 538 : 



STATIONS. 


BACK-SIGHTS. 


HEIGHT OF 
INSTRUMENT. 


FORE-SIGHTS. 


TOTAL 
HEIGHTS. 












T. P. 


INT. 




260 


4-576 


95-973 






91-397 


261 


5-420 


97-667 


3-726 




92-247 


262 








4-500 


93-167 


263 


4-910 


99-407 


3-170 




94-497 


263-40 








4-938 


94-469 


264 


3-380 


96-401 


6-386 




93-021 


265 








4640 


91-761 


266 


2-760 


93-761 


5-400 




91-001 


266-70 








3-070 


90-691 


267 








3-750 


90-011 


268 






6-925 




86-836 


+ 21-046 


—25-607 








+ 21-046 






— 4-561 








+ 91-397 






+ 86-836 



540. Third Form of Field-Book. In this the back-sights are 
placed directly under the height of the station to which they are 
taken, which lessens the chance of making mistakes in adding to 
get the height of instrument. The height of instrument is dis- 
tinguished by being included between two horizontal lines. The 
following table refers to the same ground as the preceding one : 



372 



LEVELING, 



STATIONS. 


FORE-SIGHTS. 


HEIGHTS. 


REMARKS. 


260 

261 

262 
263 

+ 40 
264 

265 

. 266 

+ 70 
267 
268 


3-726 

4-500 
3-170 

4-938 
6-386 

4-640 
5-400 

3-070 
3-750 
6-925 


91-397 
4-576 




95-973 


92-247 
5-420 


97-667 


93-167 

94-497 

4-910 


99-407 


94-469 
93-021 

3-380 


96-401 


91-761 

91-001 

2-760 


93-761 


90-691 
90-011 
86-836 



541. Best Length of Sights. There are two classes of inaccu- 
racies. With very long sights, the errors of imperfect adjustment 
and curvature are greatest ; the former varying as the length, and 
the latter as the square of the length. With very short sights, and 
therefore more numerous, the errors of inaccurate sighting at the 
target are greatest. The best usual mean is from 200 feet to 300 
feet, or more if equal distances for back-sights and fore-sights to 
turning-points can be obtained. 

542. Equal Distances of Sight. They are always very desirable. 
They are most easily determined, when no stakes have been pre- 
viously set, by " stadia " cross-hairs in the telescope of the level. 



543. Datum-LeveL This is the plane of reference, from which, 
above it or below it, usually the former, the heights of all points of 
the line are reckoned. 



THE PRACTICE. 373 

It may be taken as the height of the starting-point. If the line 
descends, it is better to call the starting-point 10 feet or 100 feet 
above some imaginary plane, so that points below the starting- 
point may not have minus-signs. 

It is desirable to refer all levels in a country to some one datum. 
This is usually the surface of the sea, and, for general purposes, 
mean tide is best. Loiv-ivater mark should be the datum when the 
levelings are connected with harbor-surveys, whose soundings al- 
ways refer to low water. Higli-water mark should be used when 
the levelings relate to the drainage of a country. 

544. Bench-Marks (B. M.). These are permanent objects, nat- 
ural or artificial, whose heights above the datum are determined 
and recorded for future reference. 

Good objects are these : Pointed tops of rocks, tops of mile- 
stones, stone door-sills, tops of gate-posts or hinges, and generally 
any object not easily disturbed, and easily described and found. 

A knob made on the spreading root 
of a tree is good. A nail may be 
driven in it, and the tree "blazed" 
and marked, as in Fig. 399. A stake 
will do till frost. 

Bench-marks should be made near 
the starting-point of a line of levels ; 
near where the line crosses a road ; on 

each side of a river crossed by it ; at the top and bottom of any 
high hill passed over ; and always at every half-mile or mile. 

The precise location and description of every bench-mark should 
be noted very fully and precisely, and in such a way that an entire 
stranger could find it, with the aid of the notes. 

545. Check-Levels, or Test-Levels. No single set of levels is to 
be trusted ; but they must be tested by another set, run between 
the bench-marks (B. M.'s), though not necessarily over the same 
ground. 

A set of levels will verify themselves if they come around to the 
starting-point again. 




374 LEVELING. 

546. Limits of Precision. Errors and inaccuracies should be 
carefully distinguished. For the latter, every leveler must make a 
standard for himself, so as to be able, in testing his work, to dis- 
tinguish any real error from his usual inaccuracy. 

The result of four sets of levelings, in France, of from 45 to 140 
miles, averaged a difference of T V foot in 43 miles, and the greatest 
error was \ foot in 56 miles. 

A French leveler, M. Bourdaloue, contracts to level the bench- 
marks of a railroad survey to within 0*002 foot per mile, or ■£$ foot 
per 50 miles. 

In Scotland, the difference of two sets of levels of 26 miles was 
0-02 foot. 

547. Trial-Levels, or Flying-Levels. Their object is to get a 
general approximate idea of the comparative heights of a portion 
of the country, as a guide in choosing lines to be leveled more ac- 
curately. More rapidity is required, and less precision is necessary. 
The distances may be measured at the same time by stadia-hairs. 

548. Leveling for Sections. The object of this is to measure all 
the ascents and descents of the line, and the distances between the 
points at which the slope changes ; so that a section or profile of it 
can be made from the observations taken. 

The line of a railroad is usually set out by a party with compass 
or transit, who drive at every hundred feet a large stake with the 
number of the station on it, and beside it a small level -peg, even 
with the surface of the ground. On this the rod is held for the 
observations. The level-peg is set in " line," and the large stake a 
foot or two to one side. 

549. Profiles. A profile is a section of ground by a vertical 
plane or cylindrical surface,* passing through the line along which 
a profile is desired. It represents to any desired scale the heights 
and distances of the various points of aline, its ascents and de- 
scents, as seen in a side view. It is made thus : Any point on the 

* A cylindrical surface is here understood to mean that formed by a line moving 
parallel to itself along any line, instead of only a circle, as in elementary geometry. 



THE PRACTICE. 



3?5 



paper being assumed for the first station, a horizontal line is drawn 
through it ; the distance to the next station is measured along it, 
to the required scale ; at the termination of this distance a vertical 
line is drawn ; and the given height of the second station above or 
below the first is set off on this vertical line. The point thus fixed 
determines the second station, and a line joining it to the first sta- 
tion represents the slope of the ground between the two. The pro- 
cess is repeated for the next station, etc. 

But the rises and falls of a line are always very small in propor- 
tion to the distances passed over, even mountains being merely as 
the roughnesses of the rind of an orange. If the distances and the 
heights were represented on a profile to the same scale, the latter 
would be hardly visible. To make them more apparent, it is usual 
to " exaggerate the vertical scale " tenfold, or more — i. e., to make 
the representation of a foot of height ten times as great as that of a 
foot of length, as in Eig. 397, in which one inch represents one 
hundred feet for the distances, and ten feet for the heights. 

In practice, engraved profile-paper is generally used, which is 
ruled in squares or rectangles, to which any arbitrary values may 
be assigned. 

When the line leveled over is not straight, the profile, whose 
length is that of the line straightened out, will extend beyond the 
"plan" when both are on the same sheet. 

550. Cross-Levels. These show the heights of the ground on a 
line at right angles to the main line. They give "■ cross-sections " 

Fig. 400. 




of it. In the note-book they are put on the right-hand page. 
They may be taken at the same time with the other levels, or inde- 



376 



LEVELING. 



pendently. In taking cross-levels where the slopes are quite steep, 
as in mountain districts, frequent settings of the instrument are 
necessary. 

A much more rapid method is by the use of "cross-section 
rods." These are two rods, one of which is about ten or twelve 
feet long, provided with a bubble-tube near each end, so as to be 
held level, and graduated to feet, tenths, and hundredths. The 
other is simply a graduated rod. The manner of using them is 
shown in Fig. 400. 

A slope-level is sometimes used. (See "Angular Surveying.") 



DIFFICULTIES. 

551. Steep Slopes. In descending or ascending a hill, the in- 
strument and the rod should be so placed that the sight should 
strike as near as possible to the bottom of the rod on the up-hill 
side, and the top of the rod on the down-hill side. 

Try this by leveling over two screws, setting the instrument so 
that one pair of opposite plate-screws shall point in the direction of 
the line, but do not be too particular ; it is a waste of time. 

Doing this produces sights of unequal length. The rod being 
about three times as high as the instrument, the down-hill sights 
will be about double the length of the up-hill ones, as shown in 
Fig. 401. Then set to one side of the line. This is necessary on 

Fig. 401. 




slopes so steep that the rod is too near the level to be read. If this 
be impossible, keep notes of the lengths of the sights to the turn- 
ing-points, backward and forward, and as soon as possible take 



DIFFICULTIES. 



377 



sights unequal in the contrary direction till the differences of 
lengths balance the former ones. When approaching a long ascent 
or descent, make these compensations in advance. 

In leveling over a line of stakes already set, as on a railroad, at 
every 100 feet, if the line of sight strikes not quite up to one, drive 
a peg as high as you can see it, and make it a turning-point, noting 
it "'peg" in the field-book. 

In leveling across a hill or hollow, instead of setting the instru- 

Fig. 402. 




ment on the top of the hill or bottom of the hollow, time will be 
saved by the method represented in Figs. 402 and 403. 



Fig. 403. 




552. When the rod is a little too lotv, raise it alongside of a 
stake, or the body, and put the top of the rod "right " ; then meas- 
ure down from the bottom of the rod, and add it to its length. 

553. When the rod is a little too high, so that the line of sight 
strikes the peg below the bottom of the rod, measure down from 
the top of the peg, and put down the sight with a contrary sign to 
what it would have had — i. e., if a back-sight make it minus, and 
if a fore-sight make it plus. 



378 



LEYELIXG. 



554. When the rod is too near. When no figure is visible, raise 

the rod slowly till a figure comes in sight. If too near to read, 
and there is no target, use a field-book as target. If the instru- 
ment is exactly over the peg, measure up to the height of the cross- 
hairs, as given by the side-screws. 

555. "Water. A. — A pond too wide to It i sighted across. Drive 
a peg to the level of the water, on the first side, and observe its 
height, as an F. S. Then drive a peg on the other side of the 
pond, also to the surface of the water. Hold the rod on it. Set 

Fig. 404. 




up the level beyond it, and sight to it as a B. S., and put down 
the observation as if it had been taken to the first peg. 



FORE-SIGHTS. 


STATIOXS. 


HEIGHTS. 


BACK-SIGHTS. 


e 


5-0 


74-89 ) 
81-89 \ 


50-00 

4S-00 


3-00 
6-00 


53-00 
54-00 



There must be no wind in the direction of the line of level. 

B. — For leveling across a running stream. Set the two pegs in 
a line at right angles to the current, although the line to be leveled 
may cross it obliquely. 

If a profile or section of the ground under the water be re- 
quired, find the height of the surface, and measure the depths be- 
low this at a sufficient number of points, measuring the distances 
also, and put these depths down as fore-sights. 

556. A Swamp, or Marsh, This can not be treated like a pond, 
for the water may seem nearly stagnant while its surface has con- 
siderable slope, its flow being retarded by vegetation. If only 
slightly "shaky," have au observer at each end of the level. If 



DIFFICULTIES. 



370 



more so, push the legs down as far as they will go, and let both 
observers lie down on their sides. If still more " shaky," drive 
three stakes or piles, to support the legs of the tripod, and stand 
the tripod on them. 

A water-level will level itself. Use that for intermediate points 
on the swamp, and test the result by leveling around the swamp 
with the spirit-level. 

557. Underwood. If it can not be cut away, set the instrument 
on some eminence, natural or artificial. 

558. Board Fence. Run a knife-blade through one of the 
boards, and hold the rod upon it on each side of the fence, as if it 
were a peg, keeping the blade in the same horizontal position while 
the rod and instrument are taken over. 

559. A Wall. First Method, Drive a peg at the bottom of 
the wall, on the first side, and observe on it. Measure the height 
of the wall above the peg, and put this down as a B. S. Drive 
another peg on the other side of the wall ; measure down to it 
from the top of the wall, and put that down as an F. S., just as if 
the level had been set in the air at the height of the top of the 

Fig. 405. 




wall, and this B. S. and F. S. had been really taken. Set up the 
instrument beyond the wall, take a B. S. to this peg, and go on as 
usual. 



FORE-SIGHTS. 


STATIONS. 


HEIGHTS. 


BACK-SIGHTS. 


© 


3-00 

12-00 

1-00 


50 
Peg. 
Peg. 

51 


74-00 
76-00 
77-00 
78-00 


5-00 

13-00 

2-00 


79-00 
89-00 
79-00 



25 



380 LEVELING. 

Second Method. Mark where the line of sight strikes the wall ; 
measure up to the top of the wall, and put this down as an F. S., 
with a plus-sign, as in 553, where the line of sight struck below 
the top of the peg. 

On the other side of the wall, sight back to it, and mark where 
the line of sight strikes. Measure to the top of the wall, and put 
this down as a B. S., with a minus-sign, and then go on as usual. 

560. House. First try to find some place for the instrument 
from which you can see through, by opening doors or windows. 
Or, find some place in the house where you can set the instrument 
and see both ways, or hold the rod at some point inside, and look 
to it from front and back. A straight stick may be used if the rod 
can not be held upright, and the height measured on the rod. 

561. The Sun. It often causes the leveler much difficulty — 

1. By shining in the object-glass. If the instrument has a 
shade on it, draw it out. If not, shade the glass with your hand 
or hat, or set the instrument to one side of the line. 

2. By heating the level unequally in all its parts. Holding an 
umbrella oyer it will remedy this. 

3. By causing irregular refraction. Some parts of the ground 
become heated more than others, and therefore rarefy the air at 
those places. This can not be avoided nor corrected. 

562. Wind. Watch for lulls of wind, and observe then several 
times, and take the mean. The least wind is at daybreak. 

563. Idiosyncrasies. Different persons do not see things pre- 
cisely alike. Each individual may have an inaccuracy peculiar to 
himself. One may read an observation higher or lower than 
another equal in skill. Also, a person's right and left eye may 
differ. This difference in individuals is termed their "personal 
equation." 

To test the accuracy of your eye, turn the head so as to bring 
the eyes in the same vertical line, and sight to the rod held hori- 
zontally. Not 3 where the vertical hair strikes. Then turn the 



LEVELING LOG ATI OK. 



381 



head to the other side, so as to invert the position of the eyes, and 
then sight again. As before, the mean of the two readings is the 
correct one. 



564. Reciprocal Leveling. This is to be used when it is im- 
possible to set midway between the two points, and the distance 
can not be readily determined. 

Set the instrument over A, and sight to a rod at B, and note 



Fig. 406. 




reading. The difference of the reading and of the height of the 
cross-hairs gives a difference of height of A and B. Then set up 
at B, and observe to A, similarly. A new difference of height is 
obtained. The mean of these two is the correct one. 

lit. of cross-hairs above peg at A=4 , 3 / Ht. of cross-hairs above peg at B=4*9' 
Observation to B=7'0 / Observation to A=4'2' 

Diff. of height =2*7' Diff. of height =0-7' 

True difference = -J (2*7' + 0'7') = 1'7'. 

Otherwise, set the instrument at an equal distance from each 
point, as A' and B', and observe to each in turn. The mean of 
the two differences of height obtained will be the true difference, 
as before. 

LEVELING LOCATION. 

565. Its Nature. It is the converse of the general problem of 
leveling, which is to find the difference of heights of two given 
points. This consists in determining the place of a point of any 
required height above or below any given point. 

To do this, hold the rod on some point of known height above 



382 



LEVELING. 



the datum-level ; sight to it, and thus determine the height of the 
cross-hairs. Subtract from this the desired height of the required 
point, and set the target at the difference. Hold the rod at the 
place where the height is desired, and raise or lower it till the 
cross-hair bisects the target. Then the bottom of the rod is at the 
desired height. Usually, a peg is driven till its top is at the given 
height above the datum. 

566. Difficulties. If the difference of height be too much to be 
measured at one setting of the instrument, take a series of levels 
up or down to the desired point. So, too, if they be far apart ; 
and thus find a place where, the instrument having a known height 
of cross-hairs, the target can finally be set, as before. 

If the ground be so low or so high that a peg can not be set 
with its top at the required height, drive a peg till its top is just 
above the surface of the ground. Observe to the rod on it, de- 
termine its height above or below the desired point, and note this 
on a large stake driven beside it ; or, place its top a whole number 
of feet above or below the required height, and mark the difference 
on it, or on a stake beside it. 



Fig. 407. 



Fill 5 



567. Staking out Work. TVhen embankments and excavations 
are to be made for roads, etc., side-stakes are set at points in their 

intended outside edges — 
i. e., where their slopes will 
meet the surface of the 
ground ; and the height 
which the ground at those 
points is above or below the 
required height or depth of 
the top or bottom of the 
finished work, is marked on 
these stakes with the words "cut," or "fill," or the signs + or — . 
The places of the stakes are found by trial. (See Gillespie's 
" Koad-Making," page 145.) These stakes are set to prepare the 
work for contractors. When the work is nearly finished, other 
stakes are set at the exact required height. 




lai \i 



LEVELING LOCATION. 



383 



Fig. 408. 



In staking out foundation-pits, set temporary stakes exactly 
above the intended bottom angles of the completed pit, thus mark- 
ing out on the surface of the 
ground its intended shape. 
Take the heights of each of 
these stakes and move them 
outward such distances that 
cutting down from them 
with the proper depth and 
slope will bring you to the 
desired bottom angle. 




568. To locate a Level-Line. This consists in determining on 
the surface of the ground a series of points which are at the same 
level — i. e., at the same height above some datum. Set one peg at 
the desired height, as in Art. 565. Sight to the rod held thereon, 
and make fast the target when bisected. Then send on the rod in 
the desired direction, and have it moved up or down along the slope 
of the ground, until the target is again bisected. This gives a sec- 
ond point. So go on as far as sights can be correctly taken, keeping 
unchanged the instrument and target. Make the last point sighted 
to a "turning-point." Carry the instrument beyond it, set up 
again, take a B. S., and proceed as at first. 

The rod should be held and pegs driven at points so near to- 
gether that the level-line between them will be approximately 
straight. 



569. Applications. One use of this operation is to mark out 
the line which will be the edge of the water of a pond to be formed 
by a dam. In that case, a point of a height equal to that of the 
top of the proposed dam, plus the height which the water will 
stand on it (to be determined by hydraulic formulas), will be the 
starting-point. Then proceed to set stakes as directed in the last 
article. 

The line from stake to stake may then be surveyed like the 
sides of a field, and the area to be overflowed thus determined. 

Strictly, the surface of the water behind a dam is not level, 



384 LEVELING. 

but is curved concavely upward, and is therefore higher and sets back 
farther than if level. The backing up of the water is called Remous. 
Another important application of this problem is to obtain 
" contour-lines " for topography. 

570. To run a Grade-Line. This consists in setting a series of 
pegs so that their tops shall be points in a line which shall have 
any required slope, ascending or descending. 

When a grade-line is to be run straight between two given 
points, set the level over one point, set the target at the height of 
the cross-hairs, hold the rod on the other point, and raise or lower 
one end of the instrument till the cross-hair' bisects the target. 
Then send the rod along the line, and drive pegs to such heights 
that when the rod is held on them the cross-hair will bisect the 
target. A stake may be driven at the extreme point to the height 
of the target. 

Another Method. Knowing the horizontal distance between 

the two given points, 
FlG - 409 - and their difference of 

level, determine the rise 



or fall per hundred feet. 
Then drive stakes at 
every hundred feet, so 
that the top of each suc- 
ceeding one is the given 
grade per hundred feet higher or lower, according as the grade 
is ascending or descending. 

For example, suppose the horizontal distance from A to B is 
1,200 feet, and that B is 16 -8 feet higher than A. The rise per 
hundred feet from A is 1 -4 foot. Beginning at A, set stakes at 
every hundred feet, so that the top of each one is 1*4 foot higher 
than the preceding one. 

A line of uniform grade or slope is not a straight line. Calling 
the globe spherical, this line, when traced in the plane of a great 
circle, would be a logarithmic spiral. On a length of six miles, the 
difference in the middle between it and its straight chord would be 
six feet. 




CHAPTER II. 



INDIRECT LEVELING. 



Fig. 410. 

9 



METHODS AND INSTRUMENTS. 

571. Vertical Surveying. Leveling may be named Vertical 
Surveying, or Up-and-dotvn Surveying ; Land-Surveying being 
Horizontal Surveying, or Right-and-left and Fore-and-aft Sur- 
veying. 

All the methods of determining the position of a point in hori- 
zontal surveying may be used in vertical surveying. 

The point may be determined by co-ordinates situated in a 
vertical plane, as in any of the systems em- 
ployed in a horizontal plane. 

Thus, if a balloon be held down by a single 
rope attached to a point in a level surface, its 
height above that surface is found by measur- 
ing the length of the rope. This is the direct 
method. It resembles that of "rectangular co-ordinates," though 
here only one of the co-ordinates is measured. 
The other might be situated anywhere in the 
surface. 

If, however, the balloon be held down by 
two cords, its height can be determined by 
measuring the length of the cords and the dis- 
tance between their lower ends. They corre- 
spond to the "focal co-ordinates." The re- 
quired vertical height can be calculated by trig- 
onometry. So in the following other indirect 
methods : 



Fig. 411. 




Fig. 412. 




386 



LEVELING. 



Fig. 413. 



The length of the string of a kite, and the angle which this 
string makes with the horizon, are the " polar co-ordinates" of the 
kite. 

The "angles of elevation" of a meteor, observed by two per- 
sons in the same vertical plane with it, and 
at known distances apart, are its "angular 
co-ordinates." 

Finally, an aeronaut could determine his 
own height by observing the angles sub- 
tended by three given objects situated on 
the earth's surface, at known distances, 
and in the same vertical plane with him. 
These would be " trilinear co-ordinates." 

Many other systems of co-ordinate lines 
and angles, variously combined, may be 
employed. 

The desired heights may also be determined by various other 
methods, analogous to those given for "inaccessible distances." 

Combinations of measurements not in the same vertical plane 
may also be used, as will be shown in this chapter. 



Fig. 414. 



572. Vertical Angles. The vertical angles measured may be 
those made — either with a level line, 
or with a vertical line — by the line 
passing from one point to the other. 
The angle B AC is called an "an- 
gle of elevation," and the angle B' A C 
an " angle of depression." The 
former angle may be called positive, 
and the latter negative. 

The angle B A Z or B' A Z is called 
the zenith-distance of the object. It 
is the complement of the former an- 
gle— i. e., = 90° — that angle taken with its proper algebraic sign. 
An angle of elevation, B A C = 10°, would be a zenith-distance of 
80°. An angle of depression, B'AC = - 10°, would be a zenith- 
distance of 100°. The zenith-distance is preferable in important 




METHODS AND INSTRUMENTS. 



387 



and complicated operations, as avoiding the ambiguity of the other 
mode of notation. 

573. Instruments. All contain a divided circle, or arc, placed 
vertically, and a level or plumb line. By these is measured the 
desired vertical angle made by the inclined line with either a 
level line or vertical line. 

This inclined line may be an actual line or a visual line. In 

the former case, it may 

, -, -, Fig. 41 6, 

be a rod, or cord, or 

wire, as shown in Figs. 
416-418. 

This last arrange- 
ment of a cord or wire 
(Fig. 418) is used in 
mine - surveying. A 
light surveyor's chain 

may be similarly used, with the advantage of giving, at the same 
time, difference of heights and distance. 




Fig. 417. 



Fig. 418. 





Difference of heights = length of chain X sin. angle. 
Horizontal distance = length of chain X cos. angle. 
These instruments are all "slope-measurers." They are also 
called Clinometers, Clisimeters, Eclimeters, etc., all meaning the 
same thing. 



574. Slopes. These may be designated by their angles with the 
horizon, or by the relations of their bases and heights. The French 
engineers name a slope by the ratio of its height to its base— i. e., 



388 



LEVELING. 



Fig. 419. 



BC 




Fig. 420. 



-r-p- ; which is the tangent of the angle 

B A C. The English and Americans use the 

AC 

ratio of the base to the height— i. e., ^-^, 

and make the height the unit, so that if 
A C = 2 C B, the slope is called 2 to 1 ; and so on. 

When the inclined line is a visual line, such as the line of 
sight of a telescope, whose angular movements are measured on a 
vertical circle beside it, and when with these is combined a hori- 
zontal circle for measur- 
ing horizontal angles, the 
instrument is called a 
"transit." 



575. Angular Profiles 
A section or profile of a 
tolerably uniform slope is 
most easily obtained, as 
shown in the figures, by 
measuring the heights or 
depths below an inclined 
line, instead of below a level line. 

A cross-section for a road may be taken in the same way. 

576. Burnier's Level. It is a pear-shaped instrument, having 
two graduated circles : one vertical, having a weight attached so as 

to keep it in the same vertical posi- 
tion when in use ; and the other, a 
horizontal graduated circle, made 
light and carried around by a mag- 
netic needle, so that the instrument 
can be used as a compass as well as 
a slope or angular level. It has a 
convex-glass, or lens, in the smaller 
end, through which can be seen a hair which covers, on the circle, 
the number of the degrees of the angle of inclination, or of the 
horizontal angle. 




Fig. 421. 




SIMPLE ANGULAR LEVELING. 



389 



The sights are on the top or sides, according as it is used as a 
compass or slope-measurer. It is used by sighting to the object, 
and at the same time reading off the angle, the hair covering the 
zero-mark when the instrument is level. 



Fig. 422. 



<rn> 



577. German Universal Instrument. Its use is to enable the 
observer to sight to an object nearly or 
quite overhead. It consists of a telescope 
having the part which carries the eye- 
piece at right angles to the part carrying 
the object-glass, instead of being in the 
same straight line, as in an ordinary tele- ^ 
scope. The part containing the eye-piece 

is connected with the other part at the axis, and is in the same 
line with the axis. 

In the telescope is placed a small mirror, or reflector, or (what 
is still better) a triangular prism of glass, at an angle of 45° to the 
line of sight. Thus the observer can keep his eye at the same place 
at any inclination of the telescope. 



SIMPLE ANGULAR LEVELING. 



578. Principle. 



Fig. 423. 



A. Foe Shoet Distances. 
For short distances, curvature and refraction 
may be neglected. Thus, if the height 
of a wall, house, tree, etc., be desired, 
note the point where the horizontal 
line strikes the wall, etc., and add its 
height above the ground to that calcu- 
lated by the formula : 
BC = AC. tang. B A C. . . . [1.] 



579. The " best-condition" angle 
for observation is 45°. Hence, in setting the instrument, we 
should, where practicable, have the distance about equal to the 
height of the point whose height we wish to ascertain. 




390 



LEVELING. 



Fig. 424. 



B. Foe Geeatee Distances. 

580. Correction for Curvature. A C is the line of apparent 
level, as given by the instrument, and A 0' 
is the line of true level. Galling the angle 
A B = 90° (which it is approximately for 
moderately great distances), formula [1] gives 
B C as the height of B above A. But B 0' 
is the true difference of heights of A and B. 
A correction for the curvature of the 
earth must therefore be made. It may be 
done in two ways : either by calculating 
CC, and adding it to B C, obtained by for- 
mula [1], or by calculating the angle C A C, 
adding it to B A C, and then applying the 
formula [1] to the angle BAC. 

581. Correcting the Result. Expressing 
the distance by *, we have, by Art. 497 : 




In feet 0' = ^ = 



= -0000000239097c; 2 . 



2 R 2 x 20912405 
Then, calling A C B a right angle, we have : 
BC' = ^X tang. B A C + 0-000000023909^ in feet. . . [2.] 
The arc A C and the straight lines A C and A C are all three 
approximately equal. 



582. Correcting the Angle. The angle CAO' = }AOC, the 
central angle, which is measured by the arc AC, or h. 
The length of the arc subtending one minute 



2 7T X 20912405 



= 6083 feet. 



360 X 60 

Then for any arc, Jc, the angle O in minutes 
h 



6083 



= 0-00016438* 



and the angle CAC (in minutes) = 0-000082193*. 

Adding this to the observed angle, B A C, and calling AC'Ba 
right angle, we have, by [1] : 

BC' = *tang. (BAC + 0-000082193*). . . . [3.] 



SIMPLE ANGULAR LEVELING. 



391 



Fig. 425. 




583. Correction for Refraction. The effect of refraction causes 
the angle actually observed to be, not CAB, but C A B', which 
will be designated by a°. For 
small distances, B and B' sensibly 
coincide. The correction for re- 
fraction may be made in two 
ways, as for curvature. 

To correct the result ly find- 
ing B B'. It varies very irregu- 
larly, with wind, barometer, tem- 
perature, etc. ; but is usually tak- 
en, as an average, BB' = 0*16 C C. 

Subtracting this from the 
value of B C, in formula [2], it 
becomes B C = h . tang. B'AC + 0-000000022F [4.] 

To correct the observed angle. Subtract from it the angle 
BAB', which is about 016 of the angle CAC. 

This changes formula [3] to 

BC' = L tang. (B'AC+ •000068447c) [5.] 

C. For Very Great Distances. 

584. Correction for Curvature. As before, there are two meth- 
ods of making the correction. 

For these distances we can not consider the angle at C a right 
angle. The triangle ABO gives 

-d ~ 7 sin. B A C 
sin. B 
To find the angle B, we have, in the triangle B A O, 
B = 180°-(O + BAO), 
B = 180° - (O + 90° + B A C), 
B= 90° -(O + BAO) ; 
Hence, sin. B = cos. (O + B A C). 
sin. BAO 



Then, B C = h . 



cos. (O+BAC)' 
sin. B A C 



and B C = B C + C C'= h . - ' , -% \ n , + 0-000000023909F. 

cos. (O + BAC) 



BC' = & 



sin. B A C 



cos. (BAC + 0-0001646&) 



0-000000023909^ 3 



[6.] 



392 



LEVELING. 



Correcting the Angle. In the triangle ABC, getting expres- 
sions for the angles, and using the sine proportion, as before, in 
ABC, we have : 

sin. (B AC + |0) 
cos. (B A C + 0) ' 
sin. (BAC + Q-Q000821931) 
cos. (BAG + 0-0001643871) ' 



B C = 1 



BC 



[7.] 



585. Correction for Refraction. Formula [6] becomes 



BC' = L 



sin. (B'AC- 0-000013751-) 



cos. (B:AC+- 0>000150636^ + Q ' Q00QQQQ23909 ^- ^ 
Formula [7] becomes, diminishing B A C in both numerator 
and denominator by 0'08 of O, 

sin. (B'AC + 0-0000684421) 
' cos. (B'A C + 0-0001506361) ' ' 



BC 



[9.] 



Fig. 426. 



586. Reciprocal Observations for canceling Refraction. Observe 

the reciprocal zenith-distances from 
each point to the other. Call these 
angles A and A'. 

The angle Z A B is the observed 
zenith-distance (A) of /?, plus the re- 
fraction p — i. e., Z A B = A + p, and 
Z'BA = A' + p\ 

Let 8 = A + p and 3' = A' + //, 
Then 8 + 8' = A + A' + p + p = ISO 
+ 0. 
The observations should be simul- 
taneous as well as reciprocal. 
When this is the case, we may take p = p. 
. Then p = 90 + |-O-i-(A + A'), 
8' = A' + p = 90 + J O + i (A'- A), 
ZAC' = 90 + iO. 
In the triangle B A C, B C : A C (= h) : : sin. B A C : sin. A B C. 
sin. B AC _ sin. (ZAB + CAO) 
•'' "^ "sin. ABC ~ yl sin. Z'BA 




BC 



sin. [180°-|(A ; - A)] 
sin. [90 + JO +i (A'-A)] s 



SIMPLE ANGULAR LEVELING. 393 



COS. I (A' — A + 0) 



When the angle is very small compared with the other angles, 
this becomes : BC' = L tan. £ (A' — A). 

Or, using angles of elevation and depression (a and /?) we haye : 

bo'=*. si °-* ( :tg m [io.] 

COS. -J (a + /3 + 0) L J 

Note.— Angle O, in minutes = 0*0001 6438*7 k. 
Log. 0-000164387 =~4-2158699. 

When is very small, compared with the other angles, by 
neglecting it we have : 

BC' = L tang. \{*+P) [11.] 

The following is from the "New York State Survey Keport," 
1882 : 

The formula employed in deducing differences of height from reciprocal 
zenith-distance observations is 

2 V 2r / ' 

where H' and H are the heights of the stations above sea-level, K is the dis- 
tance between the stations in metres, as given by the triangulation, and con- 
sequently reduced to sea-level, 7J and Z are the observed zenith-distances ; r 
is the mean radius of the earth in metres ; its logarithm is 6*80454 for lati- 
tude 43°, according to Bessel's determination. This mean value may be 
safely taken as constant throughout the area of New York State without any 
practical error in the resulting differences of height. 

/ TT 4- TT'\ 
The factor f 1 -1 J will never in this State affect H 7 — H by more 

than j^Vo P ai *t °f its value; it is usual, therefore, to compute the difference 

2' z 

of height from the formula H' — H = K tan. — - — ; and if by inspection 

2 

of a short table of values of the omitted factor it is seen that its effect will 
be appreciable, it is then introduced. 

For computing differences of height from zenith-distances observed at one 
station only, the formula 

H» - H = K cot. Z ( 1 + ! + £') + L=lH jp 

\ 2r / 2 r 

is employed. The symbols here have the same significance as before, and 
2 m is the ratio of the radius of the earth to, the radius of the curve of light. 
The value of m may be approximately determined by means of reciprocal 
zenith-distance observations. From 137 of such observations the State Sur- 
vey has found m = 0*0730 ; its value is liable to considerable fluctuation, but 
it may be considered constant within the hours to which the observations 
are confined on the survey without any material error. 



394 



LEVELING. 



(TT I ■□/ 
1 + — ) is treated as before. The logarithm of the 

coefficient ~" m is 2 . 82589. The quantity * ~ 2m K 2 has been tabulated 
Z r 2 r 

for values of K up to 18,000 metres for office use. 



Fig. 427. 



587. Reduction to the Summits of the Signals. Stations a 
and b can not be seen from each other. 
Signals are erected at each point, and 
from a the angle B a C = A is observed ; 
and from b the angle A b J) = B. The 
heights of the signals above the instru- 
ment at a and b are h and h'. The dis- 
tance between the signals is k. 

Required the reduced angles a = cab 
and f3 = D b a. 

li . cos. A 




a= A 



P=B + 



h . sin. 1" 
h' . cos. B 



[12.] 



k . sin. 1" 
The difference is in seconds. 

Usually, in such cases, zenith-distances are taken, and the ob- 
served angles are called A and A'. The reduced angles are 8 
and 8'. 

Draw a line in the figure from A to B. 
A B a we have : 

sin. A B a : sin. A : : li : 
li sin. A 



Then in the triangle 



Jc. 



or, sin. A B a = 
and a B A = 



ksm.r> 
h sin. A' 
k sin. V ' 



= A-f 



h . sin. A 



and 8' = A' -f 



li' . sin. A' 
k . sin. 1" 



[13.] 



k . sin. 1" 

The difference is seconds. 

Instead of li and h', some writers use d H and d H' ; or d A and 
d A', meaning difference of height, and difference of altitude. 

For great exactness, instead of using the mean radius of the 
earth to get 0, the radius at the point of observation is used. 



SIMPLE ANGULAR LEVELING. 



395 



Fig. 428. 




588. When the height of the signal above the instrument can 
not be measured, if the sig- 
nal be conical, like a spire, 
etc., to find B B' we meas- 
ure two diameters, 2 E 
and 2 r, and the distance 
apart, h. 

Then,BB' =i ^. [14] 

If the oblique distance 
I be measured instead of h, then 

BB '=E37^[H(E-r)] [i-(B-r)]. . . . [15.] 

When a Spire is very 
acute. From B let fall 
B I perpendicular to A B' . 
Then B I = B B' . sin. 
A B' B = A I . tan. (S" 

. _ AI.tan. (S"-S) 

sin. AB'B 
As the angle (S" — 8) is so small, 
we may take A I = A B = K. 
Now, AB'B = 180° - Z' B' A, and 
from (586) 
Z'B'A = S'=90 + |O +i(A' 

.\ Sin. A B'B = sin. {180° -[90° 
+ *0+i<A'-A)]} 
= cos. I (A'- A + 0). 
K. tan. (8' - 8) 




BB' 



cos. J (A' - A + 0) 



589. Leveling by the Horizon of the Sea. Owing to refraction, 
the apparent zenith-distance will be Z B A'. 

Let R, = radius of the earth ; HH' = horizon. 

E 



Then R + B B' = 



cos. ' 



26 



396 LEVELING. 

_. BB , =R a-co^_c) > 

cos. C L J 

Now, (1 — cos. C) = 2 sin. 2 £ C. Transposing, we have cos. 

Fig. 430. 




= cos. 2 JC — sin. 2 J C. Substituting these values in equation (1). 



we get B B' 



E(2sin. 2 i-C) 



= 2E 



sin. 2 |C 



cos. 2 $ C - sin. 2 fC " " cos. 2 i - sin. 2 J C ' 
(Developing by the binomial formula) — 

= 2Ktan. 2 |-C (1 + tan. 2 J- C - tan.^C +, etc.) 
Using the first two terms of the series, we have 
BB' = 2E tan 2 £ C (1 + tan. 2 J C). 
As the angle C is very small, we may express the tangent as an 
arc in terms of the radius, without greater error than one foot in 
an altitude of 45,000. 



E 



= ^c 2 (l + ^). [17.] 



The ande C = H B A' + A B A 



90° — n C, n being the 



coefficient of refraction. 



= 



S-90° 



1-71 

In order to introduce the value of C into equation (2), we mul- 
tiply it by the sine of 1", to reduce arc to linear measure. 



COMPOUND ANGULAR LEVELING. 



397 



Then we have 
BB'=tE( S ^)>-90T|l + i(^) 2 (S-90°)f. [18.] 

COMPOUND ANGULAR LEVELING. 

590. The following problems may mostly be reduced to a com- 
bination of : first, determining the inaccessible distance to a point 
immediately under (or over) the point whose height is desired, and 
then using this 
distance to ob- 
tain that height. 



Fig. 431. 




591. By An- 
gular Co - ordi- 
nates in one 
Plane. Take two 
stations, A and 

D, in the same vertical plane with B. At A observe the angles 
of elevation of B and D. Measure AD, At D observe the angle 
A D B. Then, in the triangle A B D we get A B, and in the tri- 
angle B A we get B C. 

sin. BDA. sin. BAC 



BC=AD. 



[19,1 



Fig. 432. 



sin. A B D .... 

For great distances, the corrections for curvature and refraction 
are to be made as in the preceding articles. 

If AD be horizontal, the same formula ap- 
plies ; but there is one angle less to measure, 
since B A C = B A D. Formula [19] gives the 
height of B above A. 

If the height of B above D, in Fig. 432, be 
desired, find BD in the triangle BAD, observe 
the angle of elevation of B from D, and then the desired height 
equals 

BD . sin. BDE. 

Otherwise, find height of D above A, and subtract it from B C. 

592. By Angular Co-ordinates in Several Planes. On irregular 
ground, when the distance between the two points is unknown, the 




398 



LEVELING. 



operations for finding it by the various methods already given 
may be combined with the observation of vertical angles, thus : 



Fig. 433. 




At A measure the vertical angle of elevation, B A C. Also meas- 
ure the horizontal angle, CAD, to some point, D, and measure 
horizontally the distance, AD. At D measure the horizontal an- 
gle, ADC. Then, 

sin. ADO 



AC = AD 



BC = AD 



sin. A C D 
sin. ADC, tang. B AC 
sin. A C D 



BC = AC. tang. B A C. 



[20.] 



593. Conversely. The distance may be obtained when the 
height is known. 

Let C B be a known height. Then, AC = C B . tan. ABC. 
B C is a known height, and D E an inaccessible line in the same 



Fig. 434. 



Fig. 435. 





horizontal plane as C. Find C D and C E by the last method, and 
measure the horizontal angle E C D subtended at C by E D. 

Then the two sides and the included angle of a triangle are 
known, to find the third side. 



CHAPTER III. 

BAROMETRIC LEVELING. 

PRINCIPLES AND FORMULAS. 

594. Principles. The difference of the heights of two places 
may be determined by finding the difference of their depths below 
the top of the atmosphere in the same way as the comparative 
heights of ground under water are determined by the difference of 
the depths below the top of the water. The desired height of the 
atmosphere above any point, such as the top of a mountain, or the 
bottom of a valley, is determined by weighing it. This is done by 
trying how high a column of mercury or other liquid the column 
of air above it will balance ; or what pressure it will exert against 
an elastic box containing a vacuum, etc. Such instruments are 
called Barometers. 

595. Applications. Since the column of mercury in the ba- 
rometer is supported by the column of air above it, the mercury 
sinks when the barometer is carried higher, and vice versa. 

The weight of any portion of air decreases from the surface of 
the earth to the assumed surface of the atmosphere. It has been 
found that, as the heights to which the barometer is carried in- 
crease in arithmetical progression, the weights of the column of air 
above the barometer, and consequently its readings, decrease in geo- 
metrical progression. Consequently, the difference of the heights 
of any two not very distant points on the earth's surface is propor- 
tional to the difference of the logarithms of the readings of the 
barometer at those points — i. e., equal to this latter difference mul- 
tiplied by some constant coefficient. This is found by experiment 
to be 60159, at the freezing-point, or temperature of 32° Fahr., the 



400 LEVELING. 

readings of the mercury being in inches, and the product, which is 
the difference of height, being in feet. 
Several corrections are necessary. 

596. Correction for Temperature of the Mercury. If the tem- 
perature of the mercury be different at the two stations, it is 
expanded at the one, and contracted at the other, to a height dif- 
ferent from that which is due to the mere weight of the air 
above it. 

Mercury expands about T o ^ 00 of its bulk for each degree of F. 
Therefore, this fraction of the height of the column is to be added 
to the height of the colder column, or subtracted from the height of 
the warmer one, in order to reduce them to the same standard. A 
thermometer is therefore attached to the instrument in such a 
manner as to give the temperature of the mercury. 

If a brass scale is used, the correction is 100 % 00 for each de- 
gree F. 

597. Correction for Temperature of the Air. The warmer the 
air is, the lighter it is ; so that a column of warm air of any height 
will weigh less than when it is colder. Consequently, the mercury 
in warm air falls less in ascending any height, and is higher at the 
place than it otherwise would be. Hence the height given by the 
preceding approximate result will be too small, and must be in- 
creased by ^l-y part for each degree F. that the temperature of the 
air is above 32°. The effect of moisture in the air changes this 
fraction to ^i-g-. 

598. Other Corrections. For very great accuracy, we should 
allow for the variation of gravity, corresponding to the variation of 
latitude on either side of the mean. So, too, we should allow for 
the decrease of gravity corresponding to any increase of height of 
the place. 

599. Rules for calculating Heights by the Mercurial Barometer. 

1. At each station read the barometer ; note its temperature by 
the attached thermometer, and note the temperature of the air by 
a detached thermometer. 

2. Multiply the height of the upper column by the difference 



PRINCIPLES AND FORMULAS. 401 

of readings of the attached thermometer, and that by tooVoo> anc ^ 
add the product to the upper column, if that be the colder, or sub- 
tract it, if that be the warmer. This gives the corrected height of 
the mercury. 

3. Multiply the difference of the logarithms of the corrected 
heights of the mercury — i. e., the corrected upper one and the 
lower one — by 60159, and the product is the approximate difference 
of heights of the places in feet for the temperature of 32°. 

4. Subtract 32° from the arithmetical mean of the tempera- 
tures of the detached thermometer ; multiply the approximate 
altitude by this difference ; divide the product by 450 ; add the 
quotient to the approximate altitude, and the sum is the cor- 
rected altitude. 

600. Formulas. The rules just given are best expressed in for- 
mulas, thus : 



Height of mercury 


AT LOWER STATION. 


AT UPPER STATION. 


H 
T 
t 


N 
V 
V 


Temperature of mercury 

Temperature of air 





Calling the reduced height of mercury at the upper station h, 

we have, by Rule 2 : 

h = 7*' + 0-00009 (T-T') h' [1.] 

N. B. — If T' is more than T, the product will be subtractive. 

Then, by Eule 3, we have : 

Approximate height = 60159 (log. 11 — log. 7i). 

By Rule 4, the correction for temperature of air 

. ,.,'.,. t + f - 64 
= approximate height X — iw\ • 

Adding this correction to the approximate height, and factor- 
ing the sum, we get : 
Corrected lit, = 60159 (log. H - log. h) (l + - + '£~ 64 ) . [2.] 

601. To correct for Latitude. Multiply the preceding result by 
0*00265 . cos. 2L (L being the latitude), and add (algebraically) 
the product to the preceding result. 



402 LEVELING. 

At 45°, correction is zero. At equator it is -f 0*00265. At 
pole it is — 0-00265. 

To correct for Elevation of the Place. Call the last corrected 
height x', and the height of the lower place above the level of the 
sea, S, and add to x' this quantity : 

x'+ 52251 S 



20912405 ' 10456203 

602. Final English Formula. Combining the previous results 
into one formula, we get : 

V 1+ 900 J' 
Ht. = 60159 (log. I - log. h){ (1 + 0-00265 . cos. 2 L), 

( s'+ 52251 _S \ 

V "*" 20912405 "*" 10456203/ 
In this formula, the three quantities under each other are three 
factors. 

Usually, only the first factor is required, and then we have for- 
mula [2]. Using the second, also, we correct for latitude ; and, 
using the third, for the elevation. 

603. French Formulas. French barometers are graduated in 
French millimetres, each = 0-03937 inch, and the thermometer 
is centigrade, in which the freezing-point is zero, and boiling-point 
100° : 

a° cent. = (frt-t- 32)° F. 
Then, the French formula corresponding to [3] is the following 
(H and li' being in millimetres, and the temperatures centigrade) : 

And the difference of heights in metres 

2 (t + f 



v ^ 1000 J' 



= 18336 (log. II - log. h) \ (1 + 0-00265 . cos. 2 L), ) [4.] 

1 + x'+ 15926\ S 



) + 



6372481 / ' 3186241 



PRINCIPLES AND FORMULAS. 403 

604. Babinet's Simplified Formula, without Logarithms. 

/ T-T\ 
h' is reduced to h, as before, viz. : h = ti ( 1 -f- 900 '). 

Then, the difference of heights in metres 



16000 . 



H - ll A i 2 ( f + O 



C+'-^r') ™ 



The heights are in millimetres and the temperatures centi- 
grade. 

Example. H = 755. h = 745 

t = 15° t' = 10°. 



Ht - = 16000 i^o( 1 + iMo)= m 



m. 



Correct result is 111*6 m. 

This formula is a very near approximation for moderate heights. 
Babinet's formula in English measures (the heights being in 
inches, and temperatures Fahrenheit) is in feet : 



H - h \ /, , t + t' — te 
Ji + 

Leslie's formula is : 



^a^) ( i + L± ^ 4 ) w 



height in feet = 55000 ?— | [7.] 

In which B = upper reading, and b = lower reading. This is 
for a temperature of 55° Fahr. 

605. Tables. These shorten the operations greatly. The most 
portable are in " Annuaire du Bureau des Longitudes." The most 
complete are Professor Guyot's, published by the Smithsonian In- 
stitution at Washington. 

606. Approximations. One tenth of an inch difference of read- 
ings in two places corresponds to about ninety feet difference of 
elevation. One millimetre difference of readings corresponds to 
about ten and a half metres difference of height, or about thirty- 
four feet. 

This is correct near the freezing-point, and near the level of the 
sea. The height corresponding to any given difference of readings 
increases, however, with the temperature and with the height of 
the station. Thus, at 70° F., -^ of an inch corresponds to an ele- 



404 



LEVELIXG. 



Fig. 436. 



yation of 95 feet ; and one millimetre at 30° cent, corresponds to 
llf metres, or about 40 feet. 

Instruments. 
607. Bako^eteks made for leveling are called Mountain Ba- 
rometers. They are either cistern barometers 
or siplion barometers. 

Fig. 436 is a cistern barometer.* This 
f consists of a column of mercury, contained in 
a glass tube, whose lower end is placed in a 
cistern of mercury. The tube is covered with 
a brass case, terminating at the top in a ring, 
A, for suspension, and at the bottom in a 
flange, B, to which the cistern is attached. 

At C is a vernier, by which the height of 
the mercury is read off. The vernier is 
moved by means of a rack, worked by the 
milled head shown at D. 

The zero of the scale is a small ivory 
point, shown below the flange B. The mer- 
cury in the cistern is raised or lowered, by 
means of the milled-headed screw 0, till its 
surface is just in contact with the ivory point. 
The upper part of the cistern is of glass, so 
that the surface of the mercury in the cis- 
tern, and the ivory point, may be readily 
seen. At E is the attached thermometer 
which indicates the temperature of the mer- 
cury. When it is carried, the mercury is 
screwed up to prevent breaking the £ - lass. 




608. The Aneroid Barometer. This is a 
thin box of corrugated copper, exhausted of 
air. When the air grows heavier, the box 
is compressed ; and when the air grows lighter, 
it is expanded by a spring inside. This motion is communicated 



* Made by Henry J. Green, 771 Broadway, New York, 



PRINCIPLES AND FORMULAS. 



405 



by suitable levers to Fig. 437. 

the index -hand, on 
the face, which indi- 
cates the pressure of 
the atmosphere, the 
face being graduated 
to correspond with a 
common barometer. 

There are several 
varieties of this in- 
strument, differing 
principally in the 
method of determin- 
ing the movement of 
the corrugated box 
due to changes in the 
density of the atmos- 
phere. 

They are made in 
sizes varying from two 

to six inches in diameter. They are much used on account of 
their portability, but are not as reliable as the mercurial barom- 
eter. 

Approximately, a difference of reading of yj^ of an inch cor- 
responds to a difference of height of nine feet. The following table 
is more nearly accurate : 




MEAN TEMPERATURE. 


30° 


40° 


50° 


60° 


70° 


80° 


Mean pressure, 27 inches 

" 28 « 

" 29 " 

30 " 


9-7 
9-3 

9-0 

8-7. 


9-9 
9-5 
9-2 

8-9 


10-1 

9-8 
9-4 
9-1 


10-3 

10-0 

9-6 

9-3 


10-5 
10-2 

98 
9-5 


10-8 
10-4 
10' 
9-7 



609. The Hypsometer. The temperature at which water boils 
varies with the pressure of the atmosphere, and therefore decreases 
in ascending heights. Then a thermometer becomes a substitute 
for a barometer. 



406 



LEVELING. 



TEMPERATURE 


CORRESPONDING 


OF BOILING WATER. 


BAROMETER READINGS. 


213° 


30"-522 


212° 


29"-922 


211° 


29"'331 


210° 


28"'751 


209° 


28 // -180 


208° 


27 /, -618 



Approximately, each degree of difference (Fahr. ) corresponds to 

about 550 feet difference of 
elevation, subject to the usual 
barometric corrections for the 
temperature of the air. (For 
minute tables, see Guyot's.) 

610. Accuracy of Baromet- 
ric Observations. This in- 
creases with the number of 
repetitions of them, the mean being taken. With great skill and 
experience they may be de- 
pended upon to a very few 
feet. 

611. The observations 
at the two places, whose 
difference of heights is to 
be determined, should be 

taken simultaneously at a series of intervals previously agreed 
upon, the distance apart of the places being as short as possible. 
Distant places should be connected by a series of intermediate ones. 



PROFESSOR GUYOT'S RESULTS. 



HEIGHTS FOUND BY THE 
BAROMETER. 


CORRESPONDING 

HEIGHTS FOUND BY THE 

SPHUT-LEVEL. 


6707 feet. 
2752 " 
6291 " 


6711 feet. 

2752 " 
\ 6285 " 
) 6293 " 



PAET III. 

TOPOGRAPHY. 



INTKODUCTION. 

612. Definition. Topography is the complete determination 
and representation of any portion of the surface of the earth, em- 
bracing the relative position and heights of its inequalities ; its 
hills and hollows, its ridges and valleys, level plains, slopes, etc., 
telling precisely where any point is, and how high it is. 

It therefore determines the three co-ordinates of any point ; 
the horizontal ones by surveying, and the vertical ones by leveling. 

The results of these determinations are represented in a conven- 
tional manner, which is called "topographical mapping." 

The difficulty is, that we see hills and hollows in elevation, 
while we have to represent them in plan. 

613. Systems. Hills are represented by various systems : 

1. By level contour-lines, or horizontal sections. 

2. By lines of greatest slope, perpendicular to the former. 

3. By shades from vertical light. 

4. By shades from oblique light. 

The most usual method is a combination of the first, second, 
and third systems. 



CHAPTER I. 



FIRST SYSTEM. 
BY HORIZONTAL CONTOUR-LINES. 

614. General Ideas. Imagine & hill to be sliced off by a num- 
ber of equidistant horizontal planes, and their intersections with it 

to be drawn as they 
would be seen from 
above, or horizontally 
projected on the map, 
as in Fig. 438. These 
are "contour-lines." 

They are the same 

lines as would be formed 

by water surrounding 

the hill, and rising one 

foot (or any other height) 

at a time till it reached 

the top of the hill. The 

edge of the water, or its 

shore, at each successive rise, would be one of these horizontal 

contour-lines. It is plain that their nearness or distance on the 

map would indicate the steepness or gentleness of the slopes. A 




Fig. 439. 



Fig. 440. 



Fig. 441. 






BY HORIZONTAL CONTOUR-LINES. 409 

right cone would thus be represented by a series of concentric 
circles, as in Fig. 439 ; an oblique cone, by circles not con- 
centric, but nearer to each other on the steep side than on the 
other, as in Fig. 440 ; and by a half-egg, somewhat as in Fig. 
441. 

615. Plane of Reference. The horizontal plane on which the 
contour-lines are projected, and to which they are referred, is 
called the " plane of reference." This plane may be assumed in 
any position, and the distance of the contour-lines above or below 
it is noted on them. It is usually best to assume the position of 
the plane of reference lower than any point to be represented ; so 
that all the contour-lines will be above it, and none of them have 



616. Vertical Distances of the Horizontal Sections. These 'de- 
pend on the object of the survey, the population of the country, 
the irregularity of the surface, and the scale of the map. In mount- 
ainous districts they may be 100 feet apart. On the United States 
Coast Survey they are twenty feet ; for engineering purposes, five 
feet, or less. One rule is to make the distance in feet equal to 
the denominator of the ratio of the scale of the map, divided 
by 600. 

617. Methods for determining Contour-Lines. They are of two 
classes : 1. Determining them on the ground at once ; 2. Deter- 
mining the highest and lowest points, and thence deducing the 
contour-lines. 

Fiest Method. 

618. General Method. Determine one point at the desired 
height of one line, and then "locate" a line at that level. 

The "reflected hand-level," or "reflecting-level," or " water- 
level," are sufficiently accurate between " bench-marks " not very 
distant. 

One such line having been determined, a point in the next 
higher or the next lower one is fixed, and the preceding operations 
repeated. 



410 



TOPOGRAPHY. 



619. On a Long, Narrow Strip of Ground, such as that required 
for locating a road : Run a section across it at every quarter or half 
mile, about in the line of greatest slope. Set stakes on these sec- 
tions at the heights of the desired contour-lines, and then get inter- 



Fig. 442. 




mediate points at these heights between the stakes. These sections 
check the levels. 

620. On a Broad Surface. Level around it setting-stakes, at 
points of the desired height, and then run sections across it, and 
from them obtain the contour-lines as before. 

The external lines here serve as checks to the cross-lines. 

621. Surveying the Contour-Lines. The contour - lines thus 
found may be surveyed by any method. If they are long, and not 
very much curved, the compass and chain and the method of "pro- 
gression " is best. If they are curved irregularly, the method of 
radiation is best. When straight lines exist among them, such as 
fences, etc., or can conveniently be established, then rectangular 
co-ordinates are most convenient. 



Second Method. 

622. General Nature. This method consists in determining the 
heights and positions of the principal points, where the surface of 
the ground changes its slope in degree or in direction — i. e., deter- 
mining all the highest and lowest points and lines, the tops of the 



BY HORIZONTAL CONTOUR-LINES. 



411 



hills and bottoms of the hollows, ridges and valleys, etc., and then, 
by proportion or interpolation, obtaining the places of the points 
which are at the same desired level. The heights of the principal 
points are found by common leveling, and their places fixed as in 
Art. 621. 

The first method is more accurate ; the second is more rapid. 



623. Irregular Ground. When the ground has no very marked 
features, run lines across it in various directions, and level along 
them, taking heights at each change of slope, just as in taking sec- 
tions for profiles. 

Otherwise, thus : Set stakes on four sides of the field, so as to 

inclose it in a rectangle, if 

•n t^v aao Fig. 443. 

possible, as in .big. 443. 

Place the stakes equidistant, 
so that the imaginary visual 
lines connecting them would 
divide the surface into rec- 
tangles. Send the rod along 
one of these lines till it gets 
in the range of a cross-one, 
and observe to it there. Put 
down the observed heights of 
these points at the corre- 
sponding points on the plat, 
on which these lines have 
been drawn. The contour-lines are determined as in Art. 626. 




624. On a Single Hill. Proceed thus : From its top, range lines 
down the hill, in various directions, and take their bearings. Set 
stakes on them at each change of slope, and note the heights and 
distances of these stakes from the starting-point, and plat their 
places. The contour-lines are then put in as in Art. 626. 

With a transit, the heights of the points could be determined 
by vertical angles ; and also their distances with stadia-hairs, their 
directions being given by the horizontal circle of the transit. The 
French use for this purpose a " leveling-compass." 
27 



412 



TOPOGRAPHY. 



625. For an Extensive Topographical Survey. Proceed thus : 
Set up and get the height of the cross-hairs from some bench-mark, 
and get the heights of high and low prominent points all around. 
Then go beyond these points and set up again. Sight to one of 
these known points as a "turning-point," and get the heights 
of all the points now in sight, as before. Then go beyond these 
again, and so on. The places of these new points are fixed as 
before. 

626. Interpolation. The heights and the places of the princi- 
pal points being determined, by either of the preceding methods, 
points of any intermediate height, corresponding to any desired 
contour-curve, are obtained by proportion. 

If, in Fig. 444, the heights of the intersection of the lines being 

found, as in Art. 623, and 
their distance apart being 
100 feet, it is required to 
construct contour - curves 
whose difference of heights 
is 5 feet : Taking, for ex- 
ample, the one whose 
height is 45 feet, we see it 
must fall between the 
points A and B, whose 
heights are 50 feet and 35 
feet ; and its distance from 
A will be found by the 
proportion, as 15 is to 5 so 
is 100 to the required distance. So on for any number of points. 
To save the labor of continually calculating the fourth propor- 
tional, a scale of proportion may be constructed. 



50.00 



Fig. 444. 

45.00 40.00 



35.00 



— ^ 


36.00 ^ 


A 

33 00 


35.00 


~-~-^____ 


-*^ 




^__ 




-^ 




\ 


^ 


' 


\ 



627. Interpolating with the Sector. This is one of the easiest 
ways. The problem is : having given on a plat two points of 
known height, to interpolate between them a point of any desired 
intermediate height. 

Take in the dividers the distance between the given points on 



BY HORIZONTAL CONTOUR-LINES. 



413 



Fig. 445. 



the plat ; open the sector so that this distance shall just reach be- 
tween numbers, on the scale marked L, corresponding to the dif- 
ference of the heights of the two 
given points — i. e., from 6 to 6, or 7 
to 7, and so on. The sector is then 
set for all the interpolations between 
these two points. 

Then note the difference of height 
between the desired point and one of 
the given points, and extend the di- 
viders between the corresponding num- 
bers on the scale. This opening will 
be the distance to be set off on the 
plat from the given point to the de- 
sired point. 




628. Ridges and Thalwegs. The general character of the sur- 
face of a country is given by two sets of lines : the ridge-lines, or 
water-shed lines ; and the " thahvegs" or "lowest lines of valleys." 

The former are lines which divide the w r ater falling upon them, 
and from which it passes off on contrary sides. They are the lines 
of least slope when looking along them from above downward ; and 
they are the lines of greatest slope when looking from below up- 
ward. They can therefore be readily determined by the slope-level, 
etc. They are the lines of least zenith-distances when viewed from 
either direction. 

On these lines are found all the projecting or protruding bends 
of the contour-lines, convex toward the lower ground, as shown in 
Fig. 396. 

The second set of lines, or the " thalwegs," are the converse of 
the former. They are indicated by the water-courses which follow 
them or occupy them. They are the lines of greatest slope when 
looked at from above, and of least slope when looked at from be- 
low. They are the lines of greatest zenith-distance when viewed 
from either direction. 

On these lines are the receding or re-entering points of the 
contour-curves, concave toward the lower ground. 



414 



TOPOGRAPHY. 



Fig. 446. 



The general system of the surface of a country is most easily 
characterized by putting down these two sets of lines, and marking 

the changes of slope, 
especially the begin- 
ning and the end. 

The most impor- 
tant points to be de- 
termined are : 

1. At the top and 
bottom of slopes. 

2, At the changes 
of slopes in degree. 

3. On the water-shed lines, and on the thalwegs. 

4. On "cols," or culminating points of passes. 

629. Forms of Ground. It will be found, on the inspection of a 
"contour-map" (which shows ground much more plainly to the 
eye than does the ground itself), that its infinite variety of form 
may, for the purposes of the engineer, be reduced to five : 

1. Sloping down on all sides — i. e., a hill (Fig. 447). 




Fig. 447. 



Fig. 448. 





2. Sloping up on all sides — i. e., a hollow (Fig. 448). 

3. Sloping down on three sides and up on one — i. e., a croupe. 



Fig. 449. 



Fig. 450. 



Fig. 451. 




BY HORIZONTAL CONTOUR-LINES. 415 

or shoulder, or promontory, the end of a ridge or water-shed line 
(Fig. 449) . 

4. Sloping up on three sides and down on one — i. e., a valley, 
or thalweg (Fig. 450). 

5. Sloping up on two sides and sloping down on two, alter- 
nately — i.e., a "pas," or "col," or "saddle" (Fig. 451). 

[Note. — The arrows in the figures indicate the direction in which water would run.] 

630. Sketching Ground by Contours. A valuable guide is, the 
observation that the contour-lines are perpendicular to the water- 
shed lines and thalwegs. Note especially the contour-lines at the 
bottoms of hills and ridges, and at the tops of hollows and valleys, 
putting them down, in their true relative positions and distances, 
to an estimated scale. 

On a long slope or hill, draw first the bottom contour-line, 
and the top one ; then the middle one ; and afterward interpolate 
others. Eemember that two of them can never meet, except on a 
perpendicular face ; and that, if one of them passes entirely around 
a hill or hollow, it will come back to its starting-point. Hold the 
field-book so that the lines on it have their true direction. As far 
as possible, all of the work should be done in the field with the 
ground in sight, and not trust to finishing from memory. 

631. Ambiguity. In contour-maps of ground, if the heights of 
the contour-lines are not written upon them, it may be doubtful 
which are the highest and lowest ; which are ridges and which 
valleys, etc. 

1. Numbers remove this. 

2. The water-courses show the slopes. If there are none, put 
some in, in the thalwegs of a rough sketch. 

3. Put hatchings on the lower sides of the contour-lines, as if 
water were draining off. 

4. Tint the valleys and low places. 

632. Conventionalities. Sometimes the spaces between contour- 
lines are colored with tints of Indian-ink, sepia, etc., increasing in 
darkness as the depth increases. 



416 TOPOGRAPHY. 

Ground under water is commonly so represented, beginning at 
the low-water line and covering the space to the six-feet-deep con- 
tour-line with a dark shade of Indian-ink ; then a lighter shade 
from 6 to 12 ; a still lighter from 12 to 18 ; and the lightest from 
18 to 24. 

Greater depths are noted in fathoms and fractions. 

633. Applications of Contour-Lines. They have many important 
uses besides their representation of ground : 

1. To obtain vertical sections — i. e., profiles. 

2. To obtain oblique sections. 

3. To locate roads. 

4. To calculate excavation and embankment. Consider the 
contour-lines to represent sections of the mass by horizontal planes. 
Then each slice between them will have its contents equal, approxi- 
mately, to half the sum of its upper and lower surfaces multiplied 
by the vertical distance apart of the sections. This method is used 
to get the cubic contents of a hill to be cut away ; of a hollow to 
be filled up ; of a great reservoir in a valley, either only projected, 
or full of water, etc. 

634. Sections by Oblique Planes. This method was much used 
by the old military topographers. It is picturesque, but not pre- 
cise. The cutting-planes are parallel, and may make any angle 
with the horizon. 



CHAPTER IT. 



SECOND SYSTEM. 
BY LINES OF GREATEST SLOPE. 

635. Their Direction. It is that which water would take in 
running down a slope. They are drawn perpendicularly to the 
contour-lines, and are the " lines of greatest slope." They are 
called "hatchings." 

Fig. 452 represents Fig. 452. 

an oval hill by this 
system. 



636. Sketching 
Ground by this System. 

This is rapid and 
effective, but not pre- 
cise. In doing this, 
hold the book to cor- 
respond with your position on the ground, and always draw to- 
ward you. If at the top of a hill, begin by drawing lines from the 
bottom, and vice versa. The hatchings are guided by contour- 
lines lightly sketched in. 




637. Details of Hatchings. They must be drawn very truly per- 
pendicular to the contour-lines. But if the contour-lines are not 
parallel, the hatchings must curve. In finishing drawings, sketch 
in the curved hatchings with a pencil at some distance apart as 
guides. When the contours are very far apart, as on nearly level 
ground, pencil in intermediate ones. 

Hatchings in adjoining rows should not be continuous, but 



418 TOPOGRAPHY. 

"break joints/' to indicate the places of the contour-lines, which 
are usually penciled in to guide the hatchings, and then rubbed 
out. The rows of hatchings must neither overlap nor separate, and 
the lines should be made slightly tremulous. "When they are put 
in without contour-lines to guide them, take care never to let two 
rows run into one ; for the breaks between the rows represent con- 
tour-lines, and two contour-lines of different heights can never 
meet except on a vertical surface. 

In drawing a hill begin at the top. "When hatchings diverge 
very much, as on hill-tops, put in alternate short ones. When the 
formation is very convex or concave, short auxiliary contours may 
be used. 



CHAPTER III. 



THIRD SYSTEM, 



IS" 



a 



Fig. 453. 

I!! 1 ! 11 ! 1 ! 1 

B 



BY SHADES FROM VERTICAL LIGHT. 

638. Degree of Shade. The steeper the slope is, the less light 
it receives, in the inverse ratio of its length — i. e., inversely as the 
secant of the angle a which it makes with 
the horizon, or directly as cos. a. Then the 
ratio of the black to the white is, 
: : 1 — cos. a : cos. a. 

In practice, the difference of shade is 
much exaggerated. 

Tables have been prepared by various na- 
tions, establishing the ratio of black and 
white. 

The proper degree of shade may be given to the hills and hol- 
lows on the map by various means. 




I'll 

! 



639. Shades by Tints. Indian-ink, or sepia, is used. The 
shades are put on with proper darkness, according to a previously 
prepared " diapason of tints." The tints are made light for gentle 
slopes, and dark for steep slopes, in a constant ratio, a slope of 60° 
being quite black, one of 30° a tint midway between that and 
white, and so on. The edges at the top and bottom are softened 
off with a clean brush. This is rapid and effective, but not very 
definite or precise, except in combination with contour-lines. 



640. Shades by Contour-Lines. This is done by making the con- 
tour-lines more numerous — i. e., interpolating new ones between 



420 



TOPOGRAPHY. 



those first determined. One objection to this is confusion of these 
lines with roads. 

641. Shades by Lines of Greatest Slope. The lines of steepest 
slope — i. e., the hatchings between the contours — have their 
thickness and distance apart made proportional to the steepness 
of the slope, in some definite ratio. This is the most usual 
method. 

The tints may be produced by varying the thickness of the 
hatchings, or their distance apart. Both are usually combined. 

642. The French Method. In this the degree of inclination is 
indicated by varying the distances between the centers of the hatch- 
ings. The rule is : the distance 'between the centers of the lines 
shall equal yfg- of an inch, plus \ of the denominator of the fraction 
denoting the declivity — i. e., tangent of the angle made by the sur- 
face of the ground ivith the plane of reference — expressed in hun- 
dredths of an inch. 

The lines are made heavier as the slope is steeper, being 
fine for the most gentle slopes, and increasing in breadth 
till the blank space between them equals \ the breadth of the 
lines. 

Only slopes of from \ to -£ T inclusive are represented by this 
method. 

643. The German, or Lehmann's Method. He uses nine grades 
for slopes from 0° to 45°, the first being white and the last black. 



Fig. 454. 



ii!'.:iii' ; : ; ,.i 



V _iil 



10° 15° 20 c 25 c 30° 




BY SHADES FROM VERTICAL LIGHT. 



421 



For the intermediate slopes, he makes the white to the black in the 
following proportion : 

The ivhite : the Hack : : 45° — angle of slope : angle of slope. 

For example, for 30° : 

light : shade : : 45° — 30° : 30° : : 1 : 2. 

Hence, the space between the strokes is to their thickness, as 
45° minus the angle of the slope is to the angle of the slope. Slopes 




steeper than 45° are represented by short, heavy lines, parallel to 
the contour-lines, as shown in the upper right-hand corner of Fig, 
455 — a hill drawn by Lehmann's method. 



644. Another Diapason of Tints : 



422 



TOPOGRAPHY. 



Slope 


2|° 


5° 


10° 


15° 


25° 


35° 


45° 


60° 


75° 


Black 


1 


2 


3 


4 


5 


6 


7 


8 


9 


White . 


10 


9 


8 


7 


6 


5 


4 


3 


■ 



This distinguishes gentle slopes better. It makes them darker, 
and the steeper slopes lighter, and provides for slopes beyond 45°. 
To use this standard, make it on the edge of a strip of paper, and 
apply that to the map in various parts, and draw a few lines cor- 
responding to the slope of those parts ; then fill up the intervening 
portions with suitable gradations. The angle of the slope is known 
from the map, since its tangent equals the vertical distance between 
the contours, divided by the horizontal distance. A scale can be 
made for any given vertical distance. 



FOURTH SYSTEM. 
BY SHADES PRODUCED BY OBLIGATE LIGHT. 

645. Light is supposed to fall from the upper left-hand corner, 
as in drawing an "elevation," although the map is in plan. Then 
slopes facing the light will have a light tint, and those on the op- 
posite side a dark tint. 

This is picturesque, but not precise. It gives apparent "re- 
lief " to the ground, but does not show the degree of steepness. 

The shades may be produced, as in the last method, by any 
means — tints, contours, or hatchings. 

By making a map with contour-lines, and shaded obliquely, it 
will be both effective and precise. 



CHAPTER IV. 



CONVENTIONAL SIGNS. 



646. Signs for Natural Surface. Sand is represented by fine 
dots made with the point of the pen ; gravel, by coarser dots. 
Rocks are drawn in their proper places, in irregular angular forms, 
imitating their true appearance as seen from above. The nature of 
the rocks, or the geology of the country, may be shown by applying 
the proper colors, as agreed on by geologists, to the back of the 
map, so that they may be seen by holding it up against the light, 
while they will thus not confuse the usual details. 

647. Signs for Vegetation. Woods are represented by scalloped 
circles, irregularly disposed, imitating trees seen '"in plan," and 
closer or farther apart according to the thickness of the forest 
(Fig. 456). It is usual to shade their lower and right-hand sides, 
and to represent their shadows, as in the figure, though, in strict- 
ness, this is inconsistent with the hypothesis of vertical light, 



Fig. 456. 



%*k w | 



4 U *rV4«<P t% 



a 






W5k -i 









.-&v *>- '. ■ 



*£^'m i ««»g 



Fig. 457. 






^T 



& 9fc 






% 



*<* ft 



****** ^\w 



■-^. 



^ yv* 



; %^c 



&& 
% 



%, 



Q 



**/ 



<*%*£ 



**** 






424 



TOPOGRAPHY. 



usually adopted for "hill-drawing." For pine and similar for- 
ests, the signs may have a star-like form, as in the lower part of 
Fig. 457. When it is desired to distinguish deciduous trees, they 
are represented as in the upper part of Fig. 457. Trees are 
sometimes drawn " in elevation," or sidewise, as usually seen 
(Fig. 458). This makes them more easily recognized, but is in 

Fig. 458. 



M ft #'***« 




iff 1 ' H ff 







Fig. 459. 




% 


% % 


%. 


% % % % 


% 


% % 


% 


% % 


%% 


% 


% % % 


%,% 


% % 


% 


%% 


% 


% % 


%. % 


%. 


% % 


% 


% % 




% 


% % 


a 


%% 


% % 


% 


A % 


% 


% % 




£L. 


@p Ql 


jm 


& f*. 





utter violation of the principles of mapping in horizontal projec- 
tion, though it may be defended as a pure convention. Orchards 
are represented by trees arranged in rows (Fig. 459). Bushes 
may be drawn like trees, but smaller. Fig. 460 represents trees 
and bushes intermingled. 

Grass-land is drawn with irregularly scattered groups of short 
lines, as in Fig. 461, the lines being arranged in odd numbers, and 



Fig. 460. 






• v < 






-tfttSrl 







sa**; 






^"---■'■L ;: 



: ^ 



'w^rJJ^ 



m&m* 







^vfW*^ 



Fig. 461. 





- 


■ 




-" 


y.v 


' .Mil', ..*. 
^ .Mil.** •^'■ I -■•'V.' .Otf 




_'"■'. .".J"' 




•■ ;• .^6 vv*. .<«*..- ^.. . ,-■» 




•- •." 


- - - . 


.**■ 




















• 




" - 




_ 


. 


.- - 






.- 


. 


"• Mi ««/. a!/,. ' w ,„. . 










v "■ , s.VK^-w.^..-...* '.. 








, 




. :■" 


1:...." 


•*-■ _~* 


... 

! 


' • 


...... 




*V'. ""'>. 




. 


■ 


' 


. ■."— . 





■ 


' . 




. 






., 


.-■■» 


-•«- 


.. 






'■ ■■ 






. 


- 







CONVENTIONAL SIGNS. 



425 



so that the top of each group is convex, and its bottom horizontal 
or parallel to the base of the drawing. Meadows are sometimes 
represented by pairs of diverging lines which may be regarded as 
tall blades of grass. Uncultivated land is indicated by appropri- 
ately intermingling the signs for grass-land, bushes, sand, and 
rocks. Cultivated land is shown by parallel rows of broken and 
dotted lines, as in the figure, representing furrows. In Fig, 462 is 
represented on the right cultivated land with fences, and on the 



Fig. 462. 



Fig. 463. 







.,:z,M\\ 



left, uncultivated land or "common." Crops are so temporary 
that signs for them are unnecessary, though often used. They are 
usually imitative, as for cotton, sugar, tobacco, rice, vines, hops, 



Fig. 464. 



Fig. 465. 




426 



TOPOGRAPHY. 



Fig. 466. 



Fig. 467. 







^^»W? ***** 



^■ ;i Aa g ,-4^ 






i: ^ : 



3 $$?-* if-^H i*? ^ 












etc. Gardens are drawn with circular and other beds and walks. 
Fig. 463 represents a house with grounds. 

648. Signs for Water. The sea-coast is represented hy drawing 
a line parallel to the shore, following all its windings and indenta- 
tions, and as close to it as possible ; then another parallel line a 
little more distant ; then a third still more distant, and so on, as 
in Fig. 468. If these lines are drawn from the low-tide mark, a 

Fig. 468. 




similar set may be drawn between that and the high-tide mark, 
and dots, for sand, be made over the included space. Fig. 464 
represents a sea-coast with rocks and reefs. 

Rivers have each shore treated like the sea-shore, as in Fig. 469. 

Brooks would be shown by only two lines, or one, according to 



CONVENTIONAL SIGNS. 



427 



Fig. 469. 



their magnitude. Ponds may be drawn like sea-shores, or repre- 
sented by parallel horizontal lines ruled across them. Marshes and 
swamps are represented by an irregular in- 
termingling of the preceding sign with that 
for grass and bushes. Fig. 465 represents a 
fresh-water marsh. Fig. 466 represents a 
salt marsh on the right and mud on the left. 
Fig. 467 represents osier-beds on the right, 
and mangrove on the left. 




649. Colored Topography. The conven- 
tional signs which have been described, as 
made with the pen, require much time and 
labor. Colors are generally used by the 
French as substitutes for them, and combine the advantages of 
great rapidity and effectiveness. Only three colors (besides In- 
dian-ink) are required, viz., gamboge (yellow), indigo (blue), and 
lake (scarlet) ; sepia, burnt sienna, yellow ochre, red-lead, and 
vermilion, are also sometimes used. The last three are diffi- 
cult to work with. To use these paints, moisten the end of 
a cake and rub it up with a drop of water, afterward diluting 
this to the proper tint, which should always be light and deli- 
cate. To cover any surface with a uniform flat tint, use a large 
camel's-hair or sable brush, keep it always moderately full, in- 
cline the board toward you, previously moisten the paper with 
clean water if the outline is very irregular, begin at the top of the 
surface, apply a tint across the upper part, and continue it down- 
ward, never letting the edge dry. This last is the secret of a smooth 
tint. It requires rapidity in returning to the beginning of a tint 
to continue it, and dexterity in following the outline. Marbling, 
or variegation, is produced by having a brush at each end of a 
stick, one for each color, and applying first one, and then the 
other, beside it before it dries, so that they may blend, but not 
mix, and produce an irregularly clouded appearance. Scratched 
parts of the paper may be painted over by first applying strong 
alum-water to the place. 

The conventions for colored topography, adopted by the French 

28 



428 



TOPOGRAPHY. 



military engineers, are as follows : Woods, yellow ; using gamboge 
and a very little indigo. Grass-land, green ; made of gamboge 
and indigo. Cultivated land, brown; lake, gamboge, and a 
little Indian-ink; "burnt sienna "will answer. Adjoining fields 
should be slightly varied in tint. Sometimes furrows are indicated 
by strips of various colors. Gardens are represented by small 
rectangular patches of brighter green and brown. Uncultivated 
land, marbled green and light iroiun. Brush, brabbles, etc., 
marbled green and yellow. Heath, furze, etc : , marbled green 
and pink. Vineyards, purple ; lake and indigo. Sands, a light 
brown; gamboge and lake ; "yellow ochre" will do. Lakes and 
rivers, light blue, with a darker tint on their upper and left-hand 
sides. Seas, dark blue, with a little yellow added. Marshes, the 
blue of water, with spots of grass, green, the touches all lying hori- 
zontally. Eoads, brown ; between the tints for sand and cultivated 
ground, with more Indian-ink. Hills, greenish-brown ; gamboge, 
indigo, lake, and Indian-ink. Woods may be finished up by draw- 
ing the trees and coloring them green, with touches of gamboge 
toward the light (the upper and left-hand side), and of indigo on 
the opposite side. 

650. Signs for Miscellaneous Objects. Too great a number of 
these will cause confusion. A few leading ones will be given : 



Signal of survey, 
Telegraph, 

Court-house, 

Post-office, 

Tavern, 

Blacksmith's shop, ^ 

Guicle-board, =f 

Quarry, ^* 

Grist-mill, 



A Fig. 470 



471 



Saiv-m ill, 
Wind-mill, 



Fig. 4T9 
cX " 480 



6 472 Steam-mill, ™ 481 

' 473 Furnace, 4S'~ 

< 474 Woolen-factory, %fe " ^83 

" 484 



475 Cotton-factory. 

476 Glass-works, ■ — ' 

477 Church, ■ J 

478 Graveyard, ^L 



" 485 
" 480 

" 487 



CONVENTIONAL SIGNS. 



429 



Fig. 488. 



An ordinary house is drawn in its true position and size, and 
the ridge of its roof shown, if the scale of the map is large enough. 
On a very small scale, a small shaded 
rectangle represents it. If colors 
are used, buildings of masonry are 
tinted a deep crimson (with lake), 
and those of wood with Indian-ink. 
Their lower and right-hand sides are 
drawn with heavier lines. Fences 
of stone or wood, and hedges, may 
be drawn in imitation of the reali- 
ties ; and, if desired, colored appro- 
priately. 

Mines may be represented by the 
signs of the planets, which were an- 
ciently associated with the various 
metals. The signs here given rep- 
resent respectively : 

Gold. Silver. Iron. Copper. Tin. Lead Quicksilver. 

A large black circle, ®, may be used for coal. 

Boundary-lines, of private properties, of townships, of counties, 
and of States, may be indicated by lines formed of various combi- 
nations of short lines, dots, and crosses, as below : 




Stone bridge. 
Wooden bridge. 

Suspension bridge. 
Aqueduct. 

Dam. 

Boat ferry. 
Eope ferry. 
Steam ferry. 

Ford for carriages. 

Ford for horses. 



+ + + + + + + + + + + + + + + + + + + + 



651. Scales. The scale to which a topographical map should be 
drawn depends on several considerations. The principal ones are 
these : It should be large enough to express all necessary details, 
and yet not so large as to be unwieldy. The scale should be so 
chosen that the dimensions measured on the ground can be easily 



430 



TOPOGRAPHY. 




THE PLANE-TABLE. 



431 




converted, without calculation, 
into the corresponding dimen- 
sions on the map. (See " Scales," 
Part I.) 

For specimens of topograph- 
ical drawing, see Enthoffer's 
"Topography," and "United 
States Coast and Geodetic Sur- 
vey Reports." 

THE PLANE-TABLE. 

652. The Plane-Table is in 
substance merely a drawing-board 
fixed on a tripod, so that lines 
may be drawn on it by a ruler 
placed so as to point to any ob- 
ject in sight. All its parts are 
mere additions to render this 
operation more convenient and 
precise.* 

Such an arrangement may be 
applied to any kind of " Angular 
Surveying," such as the Third 
Method, "Polar Surveying," in 
its two modifications of Radia- 
tion and Progression, and the 
Fourth Method, by Intersec- 
tions. Each of these will be suc- 
cessively explained. The instru- 
ment is very convenient for fill- 
ing in the details of a survey, 
when the principal points have 
been determined by the more 



f,f ta^>' : 



mm 



* The Plane-Table is not a Goniome- 
ter, or Angle-measurer, like the compass, 
transit, etc., but a Gonigraph, or Angle- 
drawer. 



432 TOPOGRAPHY. 

precise method of " Triangular Surveying," and can then be 
platted on the paper in advance. It has the great advantage of 
dispensing with all notes and records of the measurements, since 
they are platted as they are made. It thus saves time and les- 
sens mistakes, but is wanting in precision. 

653. The Table. It is usually a rectangular board of well-sea- 
soned pine, about twenty inches wide and thirty long. The paper to 
be drawn upon may be attached to it by drawing-pins, or by clamp- 
ing-plates fixed on its sides for that purpose, or by springs pressed 
upon it, or it maybe held between rollers at opposite sides of the 
table. Tinted paper is less dazzling in the sun. Cugnot's joint, or 
a pair of parallel plates, like those of the transit, may be used for 
connecting it with its tripod. A detached level is placed on the 
board to test its horizontally ; though a smooth ball, as a marble, 
will answer the same purpose approximately. 

A pair of sights, like those of the compass, are sometimes placed 
under the board, serving, like a " watch-telescope," to detect any 
movement of the instrument. To find what point on the lower 
side of the board is exactly under a point on the upper side, so that 
by suspending a plumb-line from the former the latter may be ex- 
actly over any desired point of ground, a large pair of "callipers," 
or dividers with curved legs, may be used, one of their points being 
placed on the upper point of the board, and their other point then 
determining the corresponding under point ; or a frame forming 
three sides of a rectangle, like a slate-frame, may be placed so that 
one end of one side of it touches the upper point, and the end of 
the corresponding side is under the table precisely below the given 
point, so that from this end a plumb-line can be dropped. A com- 
pass is sometimes attached to the table, or a detached compass, 
consisting of a needle in a narrow box (called a Declinator), is 
placed upon it, as desired. The edges of the table are sometimes 
divided into degrees, like the "Drawing-board Protractor." It 
then becomes a sort of goniometer. 

654. The Alidade. The ruler has a fiducial or feather edge, 
which may be divided into inches, tenths, etc. At each end it 



THE PLANE- TABLE. 433 

carries a sight like those of the compass. Two needles would be 
tolerable substitutes. The sights project beyond its edge so that 
their center lines shall be precisely in the same vertical plane as 
this edge, in order that the lines drawn by it may correspond to the 
lines sighted on by them. To test this, fix a needle in the board, 
place the alidade against it, sight to some near point, draw a line 
by the ruler, turn it end for end, again place it against the needle, 
again sight to the same point, and draw a new line. If it coincides 
with the former line, the above condition is satisfied. The ruler 
and sights together take the name of Alidade. If a point should 
be too high or too low to be seen with the alidade, a plumb-line, 
held between the eye and the object, will remove the difficulty. 

A telescope is sometimes substituted for the sights, being sup- 
ported above the ruler by a standard, and capable of pointing up- 
ward or downward. It admits of adjustments similar in principle 
to the second and third adjustments of the transit. 

But even without these adjustments, whether of the sights or 
of the telescope, a survey could be made which would be perfectly 
correct as to the relative position of its parts, however far the line 
of sight might be from lying in the same vertical plane as the edge 
of the ruler, or even from being parallel to it ; just as in the transit 
or theodolite the index or vernier need not to be exactly under the 
vertical hair of the telescope, since the angular deviation affects all 
the observed directions equally. 

655. The plane-table shown in Fig. 491 is one of the standard 
forms.* The table is leveled by means of three leveling-screws, 
and tested by a spirit-level on the alidade. The telescope of the 
alidade is "transit-mounted "—that is, has both ends of the axis 
supported. 

. Distances may be determined by means of stadia- wires placed in 
the telescope, and heights by means of the vertical arc. 

656. Method of Radiation. This is the simplest, though not 
the best, method of surveying with the plane-table. It is especially 

* Manufactured by Fauth & Co., Washington, D. C. 



434 



TOPOGRAPHY. 



Fig. 491. 




Plane-Table. 



THE PLANE-TABLE. 



435 



,>V 




applicable to surveying a field, as in the figure. In it and the fol- 
lowing figures, the size of the table is much exaggerated. Set the 
instrument at any conven- 
ient point, as ; level it, 
and fix a needle (having a 
head of sealing-wax) in the 
board to represent the sta- 
tion. Direct the alidade to 
any corner of the field, as 
A, the fiducial edge of the 
ruler touching the needle, 
and draw an indefinite line 
by it. Measure A, and 

set off the distance, to any desired scale, from the needle-point, 
along the line just drawn, to a. The line A is thus platted on 
the paper of the table as soon as determined in the field. Deter- 
mine and plat in the same way, OB, C, etc., to b, c, etc. Join 
a b, be, etc., and a complete plat of the field is obtained. Trees, 
houses, hills, bends of rivers, etc., may be determined in the same 
manner. The corresponding method with the compass or transit 
has been described. The table may be set at one of the angles of 
the field, if more convenient. If the alidade has a telescope, the 
method of measuring distances with a stadia may be here applied 
with great advantage. 



657. Method of Progression. Let AB C D, etc., be the line to 
be surveyed. Fix a needle at a convenient point of the plane-table, 
near a corner so as to leave room for the plat, and set up the table 
at B, the second angle of the line, so that the needle, whose point 
represents B, and which should be named b, shall be exactly over 
that station. Sight to A, pressing the fiducial edge of the ruler 
against the needle, and draw a line by it. Measure B A, and set 
off its length, to the desired scale, on the line just drawn, from b 
to a point a, representing A. Then sight to 0, draw an indefinite 
line by the ruler, and on it set off the length of B C from b to c. 
Fix the needle at c. Set up at C, the point c being over this sta- 
tion, and make the line c b of the plat coincide in direction with 



436 



TOPOGRAPHY. 



C B on the ground, by placing the edge of the ruler on c b, and 
turning the table till the sights point to B. The compass, if the 



Fig. 493. 




table have one, will facilitate this. Then sight forward from C to 
D, and fix CD, c cl on the plat, as I c was fixed. Set up at D, 
make dc coincide with DC, and proceed as before. The figure 
shows the lines drawn at each successive station. The table drawn 
at A shows how the survey might be commenced there. 

In going around a field, the work would be -proved by the last 
line "closing" at the starting-point; and, during the progress of 
the survey, by any direction, as from C to A on the ground, coin- 
ciding with the corresponding line, c a, on the plat. 

This method is substantially the same as the method of survey- 
ing a line with the transit. It requires all the points to be acces- 
sible. It is especially suited to the survey of a road, a brook, a 
winding path through woods, etc. The offsets required may often 
be sketched in by the eye with sufficient precision. 

When the paper is filled, put on a new sheet, and begin by 
fixing on it two points, such as C and D, which were on the former 
sheet, and from them proceed as before. The sheets can then be 
afterward united, so that all the points on both shall be in their 
true relative positions. 



658. Method of Intersection. This is the most usual and the 
most rapid method of using the plane-table. Set up the instru- 



THE PLANE-TABLE. 



437 



ment at any convenient point, as X in the figure, and sight to all 
the desired points, A, B, C, etc., which are visible, and draw in- 



Fig. 494. 




definite lines in their directions. Measure any line XY, Y being 
one of the points sighted to, and set off this line on the paper to 
any scale. Set up at Y, and turn the table till the line X Y on 
the paper lies in the direction of X Y, on the ground, as at in 
the last method. Sight to all the former points and draw lines in 
their directions, and the intersections of the two lines of sight to 
each point' will determine them, by the Fourth Method. Points 
on the other side of the line X Y could be determined at the same 
time. In surveying a field, one side of it may be taken for the 
base X Y. Very acute or obtuse intersections should be avoided — - 
30° and 150° should be the extreme limits. The impossibility of 
always doing this renders this method often deficient in precision. 
When the paper is filled, put on a new sheet, by fixing on it two 
known points, as in the preceding method. 



659. Method of Resection. This method (called by the French 
Recoupement) is a modification of the preceding method of inter- 
section. It requires the measurement of only one distance, but all 
the points must be accessible. Let A B be the measured distance. 
Lay it off on the paper as a h. Set the table up at B, and turn it 
till the line l a on the paper coincides with B A on the ground, as 
in the Method of Progression. Then sight to C, and draw an in- 
definite line by the ruler. Set up at C, and turn the line last 



438 



TOPOGRAPHY. 



drawn so as to point to B. Fix a needle at a on the table, place 
the alidade against the needle and turn it till it sights to A, Then 



Fig. 495. 




the point in which the edge of the ruler cuts the line drawn from 
B will be the point c on the table. Xext sight to D, and draw an 
indefinite line. Set up at D, and make the line last drawn point 
to 0. Then fix the needle at a or b, and by the alidade, as at the 
last station, get a new line back from either of them, to cut the 
last-drawn line at a point which will be d. So proceed as far as 
desired. 

660. To orient the Table.* The operation of orientation con- 
sists in placing the table at any point so that its lines shall have 
the same directions as when it was at previous stations in the same 
survey. 

With a compass this is very easily effected by turning the table 
till the needle of the attached compass, or that of the declinator, 
placed in a fixed position, points to the same degree as when at the 
previous station. 

Without a compass the table is oriented, when set at one end of 
a line previously determined, by sighting back on this line, as at C 
in the Method of Progression. 

* The French phrase, to " orient one's self," meaning to determine one's position, 
usually with respect to the four quarters of the heavens, of which the Orient is the 
leading one, well deserves naturalization in our language. 



THE PLANE-TABLE. 



439 



~^s 



To orient the table, when at a station unconnected with others, 
is more difficult. It maybe effected thus : Let ab on the table 
represent a line A B on the 
ground. Set up at A, make a b 
coincide with A B, and draw a 
line from a directed toward a 
steeple, or other conspicuous ob- 
ject, as S. Do the same at B. 
Draw a line c d, parallel to a b, 
and intercepted between a S, and 

b S. Divide a b and c d into the same number of equal parts. The 
table is then prepared. Now let there be a station, P, p on the 
table, at which the table is to be oriented. Set the table, so that 
p is oyer P, apply the edge of the ruler to p, and turn it till this 
edge cuts c d in the division corresponding to that in which it 
cuts a b. Then turn the table till the sights point to S, and the 
table will be oriented. 




661. To Find One's Place on the Ground. This problem may 
be otherwise expressed as interpolating a point in a plat. It is 

most easily performed by revers- 
ing the Method of Intersection. 
Set up the table over the station, 
in the figure, whose place on 
the plat already on the table is 
desired, and orient it, by one of 
the means described in the last 
article. Make the edge of the 
ruler pass through some point, a 
on the table, and turn it till the 
sights point to the corresponding 
station, A on the ground. Draw a line by the ruler. The desired 
point is somewhere in this line. Make the ruler pass through an- 
other point, b on the table, and make the sights point to B on the 
ground. Draw a second line, and its intersection with the first 
will be the point desired. Using in the same way would give a 
third line to prove the work. This operation may be used as a 




440 TOPOGRAPHY. 

new method of surveying with the plane-table, since any number 
of points can have their places fixed in the same manner. 

This problem may also be executed without orientation on the 
principle of trilinear surveying. Three points being given on the 
table, lay on it a piece of transparent paper, fix a needle anywhere 
on this, and with the alidade sight and draw lines.toward each of 
these three points on the ground. Then use this .paper to find the 
desired point, precisely as directed in the last sentence of Art. 720, 
page 487. 

When it is desired to set up the plane-table at some undeter- 
mined point, not connected by known lines with any other point 
in the survey, and the table can be readily only approximately 
oriented, the table may be accurately oriented and the point be de- 
termined by means of the " three-point problem." For the solution 
of this problem, and for treatise on the plane-table, see " United 
States Coast and Geodetic Survey Report/*' 1880, Appendix XIII. 

662. Inaccessible Distances. Many of the problems in Part I, 
Chapter V, can be at once solved on the ground by the plane-table, 
since it is at the same time a goniometer and a protractor. Thus, 
the Problem of Art. 385 may be solved as follows, on the princi- 
ple of the construction in the last paragraph of that article : Set 
the table at C. Mark on it a point, c', to represent C, placing c' 
vertically over C. Sight to A, B, and D, and draw corresponding 
lines from c'. Set up at D, mark any point on the line drawn from 
d toward D, and call it cT . Let cT be exactly over D, and direct 
d'c' toward C. Then sight to A and B, and draw corresponding 
lines, and their intersections with the lines before drawn toward A 
and B will fix points a' and b' . Then on the line joining a and b. 
given on the paper to represent A and B, a b being equal to A B on 
any scale, construct a figure, abed, similar to a'b'c'd', and the 
line c d thus determined will represent CD on the same scale as 
AB. 

663. Contouring with the Plane-Table. It is used to map the 
points on the contour-lines as soon as obtained, thus : Range out 
an approximately level line, and on it set equidistant stakes. At 



TEE PLANE-TABLE. 441 

these stakes range out perpendiculars to the line, and set up several 
stakes on them for the alignment of the rodman. Draw these lines 
on* the plane-table. Set up and " orient " the table on the ground. 
Send the rod along one of the perpendiculars till it comes to a 
point of the right height. Then sight to it with the alidade, and 
its edge will cut the corresponding line on the table at the correct 
place on the plat. So for the other perpendiculars. 



PART IV. 

TRIANGULAR SURVEYING 

OK 

By the Fourth Method. 



CHAPTEE I. 

PLANE SURFACES. 

664. Trian-gulae Sukyeyixg is founded on the method of 
determining the position of a point by the intersection of two known 
lines. Thus, the point P is determined by knowing the length of 
the line A B, and the angles P B A and P A B, which the lines P A 
and P B make with A B. By an extension of the principle, a field, 

a farm, or a country,- can be surveyed by 
measuring only one line, and calculating 
A all the other desired distances, which 

are made sides of a connected series of 
imaginary triangles, whose angles are 
A /_ \B carefully measured. The district sur- 
veyed is covered with a sort of network 
of such triangles, whence the name given to this kind of surveying. 
It is more commonly called " Trigonometrical Surveying," and 
sometimes " Geodesic Surveying," but improperly, since it does 
not necessarily take into account the curvature of the earth, 
though always adopted in the great surveys in which that is con- 
sidered. 

665. Outline of Operations. A base-line, as long as possible 
(five or ten miles in surveys of countries), is measured with ex- 
treme accuracy. 



PLANE SURFACES. 443 

From its extremities, angles are taken to the most distant ob- 
jects visible, such as steeples, signals on mountain-tops, etc. 

The distances to these and between these are then calculated by 
the rules of trigonometry. 

The instrument is then placed at each of these new stations, 
and angles are taken from them to still more distant stations, the 
calculated lines being used as new base-lines. 

This process is repeated and extended till the whole district is 
embraced by these " primary triangles " of as large sides as possible. 

One side of the last triangle is so located that its length can be 
obtained by measurement as well as by calculation, and the agree- 
ment of the two proves the accurac}^ of the whole work. 

Within these primary triangles, secondary or smaller triangles 
are formed, to fix the position of the minor local details, and to 
serve as starting-points for common surveys with chain and com- 
pass, etc. Tertiary triangles may also be required. 

The larger triangles are first formed, and the smaller ones based 
on them, in accordance with the important principle in all survey- 
ing operations, always to work from the whole to the parts, and 
from greater to less. 

666. Measuring a Base. Extreme accuracy in this is necessary, 
because any error in it will be multiplied in the subsequent work. 
The ground on which it is located must be smooth and nearly 
level, and its extremities must be in sight of the chief points in 
the neighborhood. Its point of beginning must be marked by a 
stone set in the ground with a bolt let into it. Over this a theodo- 
lite or transit is to be set, and the line " ranged out." The meas- 
urement may be made with chains, steel tapes, etc., or with rods. 

667. Measuring a Base with Rods. We will notice, in turn, 
their materials, supports, alignment, leveling, and contact. 

As to materials, iron, brass, and other metals, have been used, 
but are greatly lengthened and shortened by changes of tempera- 
ture. Wood is affected by moisture. Glass rods and tubes are pref- 
erable on both these accounts ; but wood is the most convenient. 
Wooden rods should be straight-grained white pine, etc., well sea- 
29 



4:4:4: TRIANGULAR SURVEYING. 

soned, baked, soaked in boiling oil, painted, and Tarnished. They 
may be trussed, or framed like a mason's plumb-line level, to pre- 
vent their bending. Ten or fifteen feet is a convenient length. 
Three are required, which may be of different colors, to prevent 
mistakes in recording. They must be very carefully compared 
with a standard measure. 

Supports must be provided for the rods, in accurate work. 
Posts, set in line at distances equal to the length of the rods, may 
be driven or sawed to a uniform line, and the rods laid on them, 
either directly or on beams a little shorter. Tripods or trestles, 
with screws in their tops to raise or lower the ends of the rods rest- 
ing on them, or blocks with three long screws passing through 
them and serving as legs, may also be used. Staves, or legs, for 
the rods have been used, these legs bearing pieces which can slide 
up and down them, and on which the rods themselves rest. 

The alignment of the rods can be effected if they are laid on 
the ground, by strings, two or three hundred feet long, stretched 
between the stakes set in the line, a notched peg being driven 
when the measurement has reached the end of one string, which is 
then taken on to the next pair of stakes ; or, if the rods rest on 
supports, by projecting points on the rods being aligned by the in- 
strument. 

The leveling of the rods can be performed with a common 
mason's level ; or their angle measured, if not horizontal, by a 
" slope-level." 

The contacts of the rods may be effected by bringing them end 
to end. The third rod must be applied to the second before the 
first has been removed, to detect any movement. The ends must 
be protected by metal, and should be rounded (with radius equal 
to length of rod), so as to touch in only one point. Eound-headed 
nails will answer tolerably. Better are small steel cylinders, hori- 
zontal on one end and vertical on the other. Sliding ends, with 
verniers, have been used. If one rod be higher than the next one, 
one must be brought to touch a plumb-line which touches the 
other, and its thickness be added. To prevent a shock from con- 
tact, the rods may be brought not quite in contact, and a wedge 
be let down between them till it touches both at known points on 



PLANE SURFACES. 445 

its graduated edges. The rods may be laid side by side, and lines 
drawn across the end of each be made to coincide or form one line. 
This is more accurate. Still better is a "visual contact/' a double 
microscope with cross-hairs being used, so placed that one tube 
bisects a dot at the end of one rod, and the other tube bisects a dot 
at the end of the next rod. The rods thus never touch. The dis- 
tance between the two sets of cross-hairs is of course to be added. 

A base could be measured over very uneven ground, or even 
water, by suspending a series of rods from a stretched rope by 
rings in which they can move, and leveling them and bringing 
them into contact as above. 

668. Measuring a Base with a Steel Tape. The tape should be 
from two hundred to five hundred feet long, furnished at one end 
with a spring-balance for determining the pull on the tape when 
measuring. It should be tested under the same conditions in which 
it is to be used — that is, supported at points from ten to twenty- 
five feet apart, and subjected to a pull of from ten to twenty 
pounds. The temperature at which the test is made should be 
noted. 

Let us suppose that the tape was tested, resting on supports 
twenty feet apart, and under a pull of fifteen pounds. 

To measure the base, drive stakes along the base-line twenty 
feet apart, and with one face in line. Drive nails in the lined face 
of the stakes at the same level, or on an even grade if the ground 
is not level. 

Set a post solidly in the ground at each tape-length along the 
line, so that the top of the post shall be at the height at which the 
tape is to be held. 

Place the tape on the nails in the stakes, or, better still, on hooks 
swinging from the nails, and apply a pull of fifteen pounds, bring- 
ing the ends of the tape over the posts. Hold the first graduation 
of the tape over the starting-point on the first post, and mark 
where the last graduation comes on the second post, by making a 
line on the head of a copper tack driven into the post, or on a piece 
of metal fastened on the top of the post. Bring the first gradua- 
tion on the tape to the mark on the second post, and mark the place 



446 TRIAXGULAR SURVEYING. 

of the last graduation on the third post. So proceed for the whole 
length of the line. 

A steel tape will expand -00000 7 of its length for each degree 
(Fahr.) of rise in temperature. The temperature should he care- 
fully noted when the measurement is made, and the proper correc- 
tion applied. 

The measurement is "best made on a still, cloudy day. 

If the measured line be on a slope, its measured length must be 
multiplied by the cosine of the angle of inclination, to reduce it to 
the horizontal distance between its extremities. 

669. Corrections of Base. If the rods were not level, their 
length must be reduced to its horizontal projection. This would 
be the square root of the difference of the squares of the length of 
the rod (or of the base) and of the height of one end above the 
other ; or the product of the same length by the cosine of the angle 
which it makes with the horizon.* 

If the rods were metallic, they would need to be corrected for 
temperature. Thus, if an iron bar expands 10 ooooo of its length 
for 1° Fahrenheit, and had been tested at 32°, and a base had been 
measured at 72° with such a bar ten feet long, and found to contain 
3,000 of them, its apparent length would be 30,000 feet, but its 
real length would be 8*4 feet more. An iron and a brass bar can 
be so combined that the difference of their expansion causes two 
points attached to their ends to remain at the same distance at all 
temperatures. Such a combination is used on the United States 
Coast Survey. 

Expansion for 1° Fahrenheit. 

Brass bar = 0-00001050903 ; 
Iron bar = 0-000006963535 ; 
Platinum = 0-0000051344 ; 
Glass =0-0000043119 ; 

White-pine = 0-0000022685. 

670. Reducing the Base to the Level of the Sea. Let A B = a 

* More precisely, A being this angle, and not more than 2° or 3", the difference 
between the inclined and horizontal lengths equals the inclined or real length multi- 
plied by the square of the minutes in A, and that by the decimal 0-00000004231. 



PLANE SURFACES. 



447 



be the measured base, and A' B' = x, 
the base reduced to the level of the 
sea, h the height of the measured base 
above the level of the sea, and r the 
radius of the earth to the level of the 
sea. Then we have : 

r + h : r : : a : x. 
r 



Fig. 499. 




r + h' 



X = 



ah 



r+h 



1 + 



a h 
r 
~h 



=~(^r 



Developing by the binomial formula, we get : 

-, etc. 



h ¥ , h 3 
a — x = a a-^4- a-. , 

T TV 



As h is very small in comparison with r, the first term of the cor- 
rection is generally sufficient. 

671. A Broken Base. When the angle C is very obtuse, the 
lines A C and C B being measured, and forming nearly a straight 



Fig. 500. 



E 



line, the length of the line A B is found thus : Naming the lines, 
as is usual in trigonometry, by small letters corresponding to the 
capital letters at the angles to which they are opposite, and letting 
K = the number of minutes in the supplement of the angle C, we 
shall have : 

AB = c = a + b- 0-000000042308 X ^?. 

a +0 

Log. 0-000000042308 = 2-6264222 - 10. 
Proof. Art. 12, Theorem III [Trigonometry, Appendix A], gives, 

^2 I J2 c 2 

cos. C = ; or c 5 = a? + V — 2 a I . cos. C. This becomes [Trig., 

2t Qj 

Art. 6], K being the supplement of C, c 2 = a? + ¥ + 2 a o . cos. K. The 



448 TRIANGULAR SURVEYING. 

series [Trig., Art. 5] for the length of a cosine gives, taking only its first two 
terms, since K is very small, cos. K = 1 — £K 2 . Hence, 

e = a" + ¥ + 2a I - abK 2 = (a + bf -abK 2 = (a + b) 2 (l - adK * ) ; 

\ (a + by/ 

whence, c=(a + i) |/(l - ( -^^) . 

Developing the quantity under the radical sign by the binomial theorem, and 
neglecting the terms after the second, it becomes 

Substituting for K minutes, K . siu. 1' [Trig., Art. 5], and performing the 
multiplication by a + &, we obtain 

e = a + b - ^ K V (Sln : 1/)2 • Now, (siD " 1T = 0-0000000423079 ; 

whence the formula, c = a + b — 0-000000042308 x ' . 

. a ■ + b 

672. Problem to interpolate a Base. Four inaccessible objects, 

A, B, C, D, being in a right line, 
and visible from only one point, E, 
it is required to determine the dis- 
tance between the middle points, 

iiSSSp^piSi^ B and C, the exterior distances, A B 

\\ / / and CD, being known. 

V c? Let A B = a, C D = b, BC = x; 

AEB = P,AEC = Q,AED = R 

Calculate an auxiliary angle, K, such that 

, aTr ±ab sin. Q . sin. (K - P) 

tansr. K = ; ^vi . — — fi — • t™ cY\ ■ 

s (a — h)- sin. P . siD. (K — Q) 

mt, • a + b t a — b 
Then is x = ^— ± 



2 2 . cos. K 

Of the two values of x, the positive one is alone to be taken. 
This problem is used when a portion of a base-line passe? over 
water, etc. 

Proof. In Fig. 501, produce AD to some point F. The exterior angles, 
EBC = A + P; ECD = A + Q; EDF = A + E, The triangle A B E 

B E sin. A • _. . . . Ar( „ . C E sin. A 

gives — = — — . The triangle ACE s^ves — = - — ■- . 

a sin. P a + x sin. Q 

BE a . sin. Q 



Dividing member by member, we get 



OE (a + x) sin. P ' 



PLANE SURFACES. 449 

BE sin. (A + R) 



In the same way the triangle BED and E D give 



ft + x sin. (R — P) ' 



„CE sin.(A + R) W1 , , BE (ft + x) sin. (R - Q) 

and — = — . Whence as before, — — = — ■ — . — - , — - — 

b sin.(R — Q) ' CE b . sin. (R - P) 

Equating these two values of the same ratio, we get 

a . sin. Q :<» + .) sin. (R -Q) ^ thmm 



(a + x) sin. P b . sin. (R — P) 

a b . sin. Q . sin. (R — P) ., , / , . „ 

— : --*- — v y = (a + sr) (J + x) = ab + (a + b)x + z Q . 

sm. P . sm. (R — Q) 

To solve this equation of the second degree, with reference to x, make 

_ 4 a ft sin. Q (sin. R — P) 

tan * ~ (a~_~^ ' sin. P (sin. R - Q) ' 

Then the first member of the preceding equation = £ . (a — ft) 2 x tan. 2 K, 

and we get x 2 + (a + b)x = I (a — by . tan 2 K — a ft, . 

and x= —I (a + ft) + 7[H«-ft) 2 . tan. 2 K - a ft + £(« + ft) 2 ], 

= - i (a + ft) ± ,/[i (a - ft) 2 . tan. 2 K + 1 (a - ft) 2 ], ' 

= — k(a + b).-±i (a-ft)V(tan. 2 K + 1), 

Or, since J (tan. 2 K + 1) = secant K = , we have x = ± 

cos. K 2 

a — ft 



2 . cos. K 

When a = ft, or when the two known parts are equal to each other, the 
above solution is indeterminate. For this case put 

, _ a ft sin. Q sin. (R — P) 

sin. P . sin. (R — Q) ' 
and the solution gives : 






= - i (a + ft) ± \/ tan. 2 K' + 

If a = ft, this becomes : 

x .= — I (a + ft) ± tan. K 7 . 

673. Base of Verification. As mentioned in Art. 665, a side of 
the last triangle is so located that it can be measured, as was the 
first base. If the measured and calculated lengths agree, this 
proves the accuracy of all the previous work of measurement and 
calculation, since the whole is a chain of which this is the last link, 
and any error in any previous part would affect the very last line, 
except by some improbable compensation. How near the agree- 
ment should be, will depend on the nicety desired and attained in 
the previous operations. Two bases, 60 miles distant, differed on 
one great English survey 28 inches ; on another, 1 inch ; and on a 
French triangulation extending over 500 miles, the difference was 
less than 2 feet. Results of equal or greater accuracy are obtained 



450 



TRIANGULAR SURVEYING. 



on the United States Coast Survey. "The Fire Island base, on 
the south side of Long Island, and the Kent Island base in Chesa- 
peake Bay, are connected by a primary triangulation. This Kent 
Island base is 5 miles and 4 tenths long, and the original Fire 
Island base is 8 miles and 7 tenths. The shortest distance between 
them is 208 miles, but the distance through the triangulation is 
320. The number of intervening triangles is 32, yet the computed 
and measured lengths of the Kent Island base exhibit a discrepancy 
no greater than 4 inches." 

674. Choice of Stations. The stations, or " trigonometrical 
points," which are to form the vertices- of the triangles, and to be 
observed to and from, must be so selected that the resulting trian- 
gles maybe "well-conditioned" — i. e., may have such sides and 
angles that a small error in any of the measured quantities will 
cause the least possible errors in the quantities calculated from 
them. The higher calculus shows that the triangles should be as 
nearly equilateral as possible. This is seldom attainable, but no 
angle should be admitted less than 30°, or more than 120°. When 
two angles only are observed, as is often the case in the secondary 




triangulation, the unobserved angle ought to be nearly a right 
angle. 



PLANE SURFACES, 
Fig. 60S. 



451 




452 



TRIANGULAR SURVEYING. 



To extend the triangulation, by continually increasing the sides 
of the triangles, without introducing " ill-conditioned " triangles, 
may be effected as in Fig. 502. A B is the measured base, C and D 
are the nearest stations. In the triangles ABC and A B D, all the 
angles being observed, and the side A B known, the other sides can 
be readily calculated. Then, in each of the triangles D A C and 
D B C, two sides and the contained angles are given to find D C, 
one calculation checking the other. D then becomes a base to 
calculate E F, which is then used to find GH, and so on. 

The fewer primary stations used the better, both to prevent 
confusion and because the smaller number of triangles makes the 
correctness of the results more "probable." 

The United States Coast and Geodetic Survey displays some 
fine illustrations of these principles, and of the modifications they 
may undergo to suit various localities. Fig. 503 represents part 
of the scheme of the primary triangulation resting on the Massa- 
chusetts base, and including some remarkably well-conditioned tri- 
angles, as well as the system of quadrilaterals, which is a valuable 
feature of the scheme when the sides of the triangles are extended 
to considerable lengths, and quadrilaterals, with both diagonals 
determined, take the place of simple triangles. 

The engraving is on a scale of 1 : 1,200,000. 



Fig. 504. 



675. Signals. They must be high, conspicuous, and so made 
that the instrument can be placed precisely under them. 

Three or four timbers framed into a 
pyramid, as in Fig. 504, with a long mast 
projecting above, fulfill the first and last 
conditions. The mast may be made verti- 
cal by directing two theodolites to it, and 
adjusting it so that their telescopes follow 
it up and down, their lines of sight being 
at right angles to each other. Guy ropes 
may be used to keep it vertical. 

Another form of signal is represented 
in the three following figures. It consists 
merely of three stout sticks, which form a tripod, framed with the 




PLANE SURFACES. 



453 



signal-staff, by a bolt passing through their ends and its middle. 
Fig. 505 represents the signal as framed on the ground ; Fig. 506 
shows it erected and ready for observation, its base being steadied 

Fig. 505. Fig. 506. Fig. 507. 






with stones ; and Fig. 507 shows it with the staff turned aside, to 
make room for the theodolite and its protecting tent. The heights 
of these signals varied between fifteen and eighty feet. 

Another good signal consists of a stout post let into the ground, 
with a mast fastened to it by a bolt below and a collar 
above. By opening the collar, the mast can be turned 
down and the theodolite set exactly under the former 
summit of the signal, i. e. , in its vertical axis. 

A tripod of gas-pipe has been used to support the 
signal in positions exposed to the sea, as on shoals. 
It is taken to the desired spot in pieces, and there 
screwed together and set up. 

Signals should have a height equal to at least T oVo 
of their distance, so as to subtend an angle of half a 
minute, which experience has shown to be the least allowable. 

To make the tops of the signal-masts conspicuous, flags may be 
attached to them : white and red, if to be seen against the ground ; 
and red and green, if to be seen against the sky.* The motion of 

* To determine at a station A, 
whether its signal can be seen from 
B, projected against the sky or not, 
measure the vertical angles B A Z 
and Z A C. If their sum equals or 
exceeds 180°, A will be thus seen 
from B. If not, the signal at A 
must be raised till this sum equals 
180°. 





454 



TRIANGULAR SURVEYING. 



flags renders them visible, when much larger motionless objects are 
not ; but they are useless in calm weather. A disk of sheet-iron, 
with a hole in it, is very conspicuous. It should be arranged so 
as to be turned to face each station. A barrel, formed of muslin 
sewed together, four or five feet long, with two hoops in it two feet 
apart, and its loose ends sewed to the signal -staff, which passes 
through it, is a cheap and good arrangement. A tuft of pine- 
boughs fastened to the top of the staff will be well seen against the 
sky. 

In sunshine a number of pieces of tin, nailed to the staff at 
different angles, will be very conspicuous. A truncated cone of 
burnished tin will reflect the sun's rays to the eye in almost every 
situation. 

The most perfect arrangement is the " heliotrope." This con- 
sists of a mirror a few inches in diameter, so mounted on a tele- 



Fig. 510. 




scope, near the eye-end, that the reflection of the sun may be 
thrown in any desired direction. They have been observed on at a 
distance of nearly two hundred miles, when the outlines of the 
mountains on which they were placed were invisible. A man, 
called a "heliotroper," is stationed at the instrument. He directs 
the telescope toward the station at which the transit is placed for 
observation, and keeps the mirror turned so as to reflect the sun in 
a direction parallel to the axis of the instrument. This he accom- 
plishes by causing the reflection to pass through two perforated 



PLANE SURFACES. 



455 



disks, mounted on the telescope, one near the object-end, and the 
other near the mirror. 

For night-signals, an Argand lamp has been used ; or, better 
still, a Drummond light, or a magnesium-light. The distinctness 
of the light is exceedingly increased by a parabolic reflector behind 
it, or a lens in front of it. 

676. Observations of the Angles. These should be repeated as 
often as possible. In extended surveys, three sets, of ten each, are 
recommended. They should be taken on different parts of the 
circle. In ordinary surveys, it is well to employ the method of 
"traversing." In long sights, the state of the atmosphere has a 
very remarkable effect on both the visibility of the signals and on 
the correctness of the observations. 

When many angles are taken from one station, it is important 
to record them by some uniform system. The form given below is 
convenient. It will be noticed that only the minutes and seconds 
of the second vernier are emplo} T ed, the degrees being all taken 
from the first : 

Observations at . 



STATIONS 


READINGS. 


MEAN 
READINGS. 


RIGHT OR LEFT 




OBSERVED 
TO. 




OF PRECEDING 
OBJECT. 


REMARKS. 


VERNIER A. 


VERNIER B. 


A 


70 19 


/ // 

18 40 


70 18 50 






B 


103 32 20 


32 40 


103 32 30 


R. 




C 


115 14 20 


14 50 


115 14 35 


R. 





When the angles are " repeated," the multiple arcs will be regis- 
tered under each other, and the mean of the seconds shown by all 
the verniers at the first and last readings be adopted. 

When the country over which the triangulation extends is flat, 
it has been found necessary to elevate the transit some distance 
from the surface of the ground, the stratum of air near the surface 
being so disturbed by exhalations and inequalities of temperature 
ami density as to render accurate observation impossible. The 
plan adopted on the Coast Survey is as follows : On the top of a 
signal-tripod, forty-three feet high, is placed a cap-block, into 
which is mortised a square hole to receive the signal-pole. Around 



456 TRIANGULAR SURVEYING. 

the tripod, but not touching it, is erected a rectangular scaffold, 
forty feet high. On the top of it is a platform, from which the 
observations are taken, the signal-pole being removed from the cap- 
block, and the transit placed so that its center shall be precisely 
over the station-point. 

677. Seduction to the Center. It is often impossible to set the 
instrument precisely at or under the signal which has been ob- 
served. In such cases proceed 
FlG - B11 - thus : Let C be the center of 

the signal, and RCL the de- 
sired angle, E being the right- 
hand object and L the left-hand 
D< ^~ — -^^^ one. Set the instrument at D, 

^^~ ;:;;;:c ^ 2 Ajt > as near as possible to C, and 

measure the angle EDL. It 
may be less than E C L, or greater than it, or equal to it, accord- 
ing as D lies without the circle passing through C, L, and E, or 
within it, or in its circumference. The instrument should be set 
as nearly as possible in this last position. To find the proper cor- 
rection for the observed angle, observe also the angle L D C (called 
the angle of direction), counting it from 0° to 360°, going from 
the left-hand object toward the left, and measure the distance 
DO. Calculate the distances CEandCL with the angle EDL, 
instead of E C L, since they are sufficiently nearly equal. Then, 
T?nr T>-n-r,CD. sin. (E D L + L D C) CD. sin. L D C 

ecl = edl + — - 0R . sin . r mnsnr- 

The last two terms will be the number of seconds to be added 
or subtracted. The trigonometrical signs of the sines must be 
attended to. The log. sin. 1" = 4-6855749. Instead of divid- 
ing by sin. 1", the correction without it, which will be a very 
small fraction, may be reduced to seconds by multiplying it by 
206265. 

Example. Let E D L = 32° 20' 18-06' ; L D C 101° 15' 32«4f : 
CD = 0-9; CE = 35845-12; CL = 29783*1. 

The first term of the correction will be + 3 -750", and the 
second term — 6 -113". Therefore, the observed angle EDL 



PLANE SURFACES. 



457 



Fig. 512. 




must be diminished by 2 , 363 // , to reduce it to the desired angle 
RCL. 

Much calculation may be saved by taking the station D so that 
all the signals to be observed can be seen from it. Then only a 
single distance and angle of direction need be meas- 
ured. 

It may also happen that the center, C, of the 
signal can not be seen from D. Thus, if the signal 
be a solid circular tower, set the theodolite at D, 
and turn its telescope so that its line of sight be- 
comes tangent to the tower at T, T' ; measure 
on these tangents equal distances, D E, DF, and 
direct the telescope to the middle, G, of the line 
E F. It will then point to the center, ; and the distance D C 
will equal the distance from D to the tower plus the radius ob- 
tained by measuring the circumference. 

If the signal be rectangular, measure D E, D F. 

Fig. 513. Take any point G on D E, and on DF set off D H 

= D G ?-f . Then is G H parallel to E F (since 

D G : D H : : D E : D F), and the telescope di- 
rected to its middle, K, will point to the middle of 
the diagonal E F. We shall also have D C = D K 
DE 
DG" 
Any such case may be solved by similar methods. 
The "pliase" of objects is the effect produced by the sun shin- 
ing on only one side of them, so that the telescope will be directed 
from a distant station to the middle of that bright side instead of 
to the true center. It is a source of error to be guarded against. 
Its effect may, however, be calculated. 
When the signal is a tin cone : 
Let r = radius of the signal ; 

Z == angle at the point of observation between the sun 

and the signal ; 
D = the distance. 




Then, the correction == ± 



r cos. 2 JZ 
DTin. 1"' 



458 TRIANGULAR SURVEYING. 

678. Correction of the Angles. When all the angles of any tri- 
angle can be observed, their sum should equal 180°.* If not, they 
must be corrected. If all the observations are considered equally 
accurate, one third of the difference of their sum from 180° is to 
be added to, or subtracted from, each of them. But if the angles 
are the means of unequal numbers of observations, their errors may 
be considered to be inversely as those numbers, and they may be 
corrected by this proportion ; As the sum of the reciprocals of each 
of the three numbers of observations is to the whole error, so is 
the reciprocal of the number of observations of one of the angles 
to its correction. Thus, if one angle was the mean of three obser- 
vations, another of four, and the third of ten, and the sum of all 
the angles was 180° 3', the first-named angle must be diminished by 
the fourth term of this proportion ; J + 4 + iV : 3' : : \ : 1' 27*8". 
The second angle must in like manner be diminished by 1' 5*9" ; 
and the third by 26*3". Their corrected sum will then be 180°. 

It is still more accurate, but laborious, to apportion the total 
error, or difference from 180°, among the angles inversely as the 
"weights." On the United States Coast Survey, in six triangles 
measured in 1844 by Professor Bache, the greatest error was six 
tenths of a second. 

678 1 . Calculation and Platting. The lengths of the sides of the 
triangles should be calculated with extreme accuracv, in two ways 
if possible, and by at least two persons. Plane trigonometry may 
be used for even large surveys ; for, though these sides are really 
arcs and not straight lines, the difference will be only one twentieth 
of a foot in a distance of 11J miles ; half a foot in 23 miles ; a foot 
in 34£ miles, etc. 

The platting is most correctly done by constructing the tri- 
angles, by means of the calculated lengths of their sides. If the 
measured angles are platted, the best method is that of chords. 
If many triangles are successively based on one another, they will 
be platted most accurately by referring all their sides to some one 

* If the triangles were very large, they would have to be regarded as spherical, 
and the sum of their angles would be more than 180°; but this "spherical excess" 
would be only 1" for a triangle containing 76 square miles, 1' for 4,500 square miles, 
etc. ; and may therefore be neglected in all ordinary surveying operations. 



PLANE SURFACES. 459 

meridian line by means of " Bectangular Co-ordinates." In the 
survey of a country, this meridian would be the true north and 
south line passing through some well-determined point. 

679. Interior Filling up. The stations whose positions have 
been determined by the triangulation are so many fixed points, 
from which more minute surveys may start and interpolate any 
other points. The trigonometrical points are like the observed lati- 
tudes and longitudes which the mariner obtains at every opportu- 
nity, so as to take a new departure from them and determine his 
course in the intervals by the less precise methods of his compass 
and log. The chief interior points may be obtained by " Second- 
ary Triangulation," and the minor details be then filled in by 
any of the methods of surveying, with chain, compass, or transit, 
already explained, or by the plane-table. With the transit, "Trav- 
ersing " is the best mode of surveying, the instrument being set at 
zero, and being then directed from one of the trigonometrical 
points to another, which line therefore becomes the "meridian " 
of that survey. On reaching this second point, in the course of the 
survey, and sighting back to the first, the reading* should of 
course be 0°. 

680. Radiating Triangulation. This name may be given to a 
method shown in the figure. Choose a conspicuous point, 0, nearly 
in the center of the field or farm to be 

surveyed. Fmd other points, A, B, C, D, 
etc. , such that the signal at can be 
seen from all of them, and that the tri- 
angles ABO, B C 0, etc., shall be as 
nearly equilateral as possible. Measure 
one side, A B for example. At A measure 
the angles A B and OAG; at B meas- 
ure the angles B A and B C ; and so 
on, around the polygon. The correctness 
of these measurements may be tested by 

the sum of the angles. It may also be tested by the trigonometri- 
cal principle that the product of the sines of every alternate angle, 
30 




460 TRIANGULAR SURVEYING. 

or the odd numbers in the figure, should equal the product of the 
sines of the remaining angles, the even numbers in the figure. 

The triangles A O B, B O C, COD, etc., give the following proportions 
[Trigonometry, Art. 12, Theorem I] : A : O B : : sin. (2) : sin. (1); OB : 
0:: sin. (4) : sin. (3) ; O C : O D : : sin. (6) : sin. 5 ; and so on around the 
polygon. Multiplying together the corresponding terms of all the propor- 
tions, the sides will all be canceled, and there will result 
1:1:: sin. (2) x sin. (4) x sin. (6) x sin. (8) x sin. (10) x sin. (12) x sin. (14) ; 
sin. (1) x sin. (8) x sin. (5) x sin. (7) x sin. (9) x sin. (11) x sin. (13). 
Hence the equality of the last two terms of the proportion. 

The calculations of the unknown sides are readily made. In 
the triangle ABO, one side and all the angles are given to find 
A and B 0. In the triangle B C 0, B and all the angles are 
given to find B and C ; and so with the rest. Another proof 
of the accuracy of the work will be given by the calculation of the 
length of the side A in the last triangle, agreeing with its length 
as obtained in the first triangle. 

681. Farm Triangulation. A farm or field may be surveyed by 
the previous methods, but the following plan will often be more 

convenient : Choose a base, as X Y, 
„ within the field, and from its ends meas- 

ure the angles between it and the direc- 
tion of each corner of the field, if the 
theodolite or transit be used, or take the 
bearing of each, if the compass be used. 
Consider first the triangles which have 
X Y for a base, and the corners of the 
field, A, B, C, etc., for vertices. In each of them one side and the 
angles will be known to find the other sides, X A, X B, etc. Then 
consider the field as made up of triangles which have their vertices 
at X. In each of them two sides and the included angle will be 
given to find its content. If Y be then taken for the common ver- 
tex, a test of the former work will be obtained. 

The operation will be somewhat simplified by taking for the 
base-line a diagonal of the field, or one of its sides. 

682. Inaccessible Areas. A field or farm may be surveyed, by 
this " Fourth Method," without entering it. Chocse a base-line 




PLANE SURFACES. 461 

XY, from which all the corners of the field can be seen. Take 
their bearings, or the angles between the base-line and their direc- 
tions. The distances from X and Y to each 
of them can be calculated as in the last 
article. The figure will then show in what 
manner the content of the field is the differ- 
ence between the contents of the triangles, 
having X (or Y) for a vertex, which lie out- 
side of it, and those which lie partly with- 
in the field and partly outside of it. Their 
contents can be calculated as in the last 
article, and their difference will be the de- 
sired content. If the figure be regarded as generated by the rev- 
olution of a line one end of which is at X, while its other end 
passes along the boundaries of the field, shortening and length- 
ening accordingly, and if those triangles generated by its movement 
in one direction be called plus and those generated by the contrary 
movement be called minus, their algebraic sum will be the content. 

683. Inversion of the Fourth Method. In all the operations 
which have been explained, the position of a point has been de- 
termined, as in Art. 6, by taking the angles, or bearings, of two 
lines passing from the two ends of a base-line to the unknown 
point. But the same determination may be effected inversely, by 
taking from the point the bearings, by compass, of the two ends of 
the base-line, or of any two known points. The unknown point 
will then be fixed by platting from the two known points the op- 
posite bearings, for it will be at the intersection of the lines thus 
determined. 

684. Defects of the Method of Intersection. The determination 
of a point by the Fourth Method, founded on the intersection of 
lines, has the serious defect that the point sighted to will be very 
indefinitely determined if the lines which fix it meet at a very acute 
or a very obtuse angle, which the relative positions of the points 
observed from and to often render unavoidable. Intersections at 
right angles should therefore be sought for, so far as other consid- 
erations will permit. 



CHAPTER II. 



SPHERICAL SURVEYING, OR GEODESY. 



Fig. 51' 



685. Nature. It comprises the methods of surveying areas of 
such extent that the curvature of the earth can not be neglected. 
The general method is the same as that given in Chapter I, but 
more precise methods of measurement and of computation are 
required, since the triangles into which the surface is divided are 
spherical triangles. 

The United States Coast and Geodetic Survey, the Lake Sur- 
vey, and the State Surveys organized by several of the States, are 
works of this character. 

The subject is too extensive to be properly treated within the 

limits of this work. Only a general 
sketch of it will be given, with ref- 
erences to such authorities as will 
enable the student to further inves- 
tigate the subject. 

Field- Work. 
686. Reconnaissance. The first 
step in making a geodetic survey is 
the selection of a series of points. 
A, B, C, etc. (Fig. 517), as the basis 
of a system of triangulation. In 
case the country is broken or open, 
but little difficulty will be experienced in locating these points, 
and often lines of great length may be secured. Thus, in the tri- 
angulation of California,* the line Mount Helena-Mount Shasta 




See " Report of Coast and Geodetic Survey," 1S68, 1876, 1880, 188$ 



SPHERICAL SURVEYING, OR GEODESY. 463 

is 192 miles in length. It is in general advisable to choose the 
points so that the resulting triangle sides are as nearly equal as 
possible. To do this, it may be necessary to build towers or scaf- 
folds at the stations A, B, etc., on which to place the instrument. 
Signals must also be placed at the stations sighted at, their general 
character depending on the length of the lines of sight. 

687. The Base. In order to compute a triangulation, we must 
have at least one side measured. This measured side is called the 
base-line, or simply the base. In geodetic work the base must be 
measured with great accuracy, though it is more important that 
many bases occur in a system, and these be measured with mod- 
erate precision, than that only a few occur, and these be measured 
with great precision. The reason is, that a check can be more fre- 
quently had of the character of the work. 

Several different forms of base-measuring apparatus * have been 
designed, of which probably the simplest and best consists of a steel 
bar packed in melting ice. The bar will remain of the same length 
throughout the measurement, as its temperature is always 32° Eahr. 

688. The Angles. Suppose the observer at any station, as D 
for example. The angles to be measured would be ADO, CD E, 
E D F. Each of these angles should be measured independently a 
number of times, depending on the quality of the instrument used, 
and the mean of the results taken. As a check against mistakes 
and accidental errors of various kinds, combinations of these an- 
gles should be measured, as AD E, A D F, C D F. On the method 
of measuring an angle with a theodolite, see Wright's " Adjust- 
ment of Observations," pp. 253, 254. 

Office-Work. 

689. Computation of the Sides of the Triangles. The triangles 
observed are supposed to have sides of such length that the sum of 

* For descriptions of various forms of base apparatus, see " Report of United 
States Coast and Geodetic Survey," 1854, 1857, 1880, 1882; "Report of Primary 
Triangulation of the United States Lake Survey " ; Wright's " Adjustment of Obser- 
vations," Chapter VII^* 



464 TRIANGULAR SURVEYING. 

the three angles exceeds 180° by a certain sensible quantity called 
the spherical excess. This is usually only a few seconds. For 
a triangle containing about 76 square miles, which, if equilateral, 
would have sides 13 miles long, the spherical excess is only one 
second. For a triangle with sides of 102 miles it is one minute. 
It must be determined before we can know how much the error 
of closure is, and therefore what the correction to each angle 
should be. 

690. Spherical Excess. Calling the earth a sphere, the spheri- 
cal excess e (in seconds) of a triangle is found from the relation 
_ area of triangle 
6 ~ ~R 2 dnTT ' 
when E = the radius of the earth. 

The triangle surface being small, compared with R 2 , may be 
obtained with sufficient accuracy by treating it as if it were plane. 
Thus, when two sides and the contained angle are given, we have : 
area = % ab sin. C ; 

, ,, „ a b sin. C 

and therefore e = _ _» . — -» . 

2 R" sin. 1 

The earth, however, instead of being spherical, is spheroidal in 
form ; and since a spheroidal triangle may be computed as a spheri- 
cal triangle on a sphere whose radius is VRN, when R aud N are 
the radii of curvature of the meridian and of the section normal to 
the meridian at the mean of the latitudes of the triangle vertices, 
we replace R 2 in the above value of e by R N". We have then : 

, a b sin C. 

excess m seconds = - _. .. T -= . 

2 R ^h arc 1 

Writing this in the form 

e = m a b sin. C, 
the values of m may be taken from the following table, the argu- 
ment being the mean latitude of the triangle vertices. The metre 
is the unit of length : 



SPHERICAL SURVEYING, OR GEODESY. 



465 



LAT. 


LOG. m. 


LAT. 


log. m. 


LAT. 


LOG. m. 


LAT. 


LOG. Ml. 


o 

10 


1-40675 


o 

25 


1-40589 


o 

40 


1-40451 


o 

55 


1-40299 


11 


1-40672 


26 


1-40581 


41 


1-40441 


56 


1-40289 


12 


1-40668 


27 


1-40573 


42 


1-40431 


57 


1 -40280 


13 


1-40663 


28 


1-40564 


43 


1-40420 


58 


1-40271 


14 


1-40659 


29 


1-40555 


44 


1-40410 


59 


1-40262 


15 


1-40654 


30 


1-40547 


45 


1-40400 


60 


1-40253 


16 


1-40649 


31 


1-40537 


46 


1-40390 


61 


1-40244 


17 


1 40643 


32 


1-40528 


47 


1-40380 


62 


1-40235 


18 


1-40637 


33 


1-40519 


48 


1-40369 


63 


1-40226 


19 


1-40631 


34 


1-40509 


49 


1-40359 


64 


1-40218 


20 


1-40625 


35 


1-40500 


50 


1-40349 


65 


1-40210 


21 


1-40618 


36 


1-40491 


51 


1-40339 


6Q 


1-40202 


22 


1-40611 


37 


1-40481 


52 


1-40329 


67 


1-40195 


23 


1-40604 


38 


1-40471 


53 


1-40319 


68 


1-40188 


24 


1-40597 


39 


1-40461 


54 


1-40309 


69 

70 


1-40181 
1-40174 



Example. In a spherical triangle, given a = 122755 , b = 
94616 m , angle C = 50° 10' 20", mean latitude of vertices, A, B, C = 
45° 15' ; required the spherical excess. 

log. a, 5-08904 

log. b, 4-97596 

log. sin. C, 9-88535 

log. m, 1-40398 

log. 22-61, 1-35433 

whence excess e — 22" '61. 

691. Having found the spherical excess, if the sum of the angles 
of the triangle is not equal to 180° plus this excess, the difference is 
distributed among them, and each angle is corrected by one third 
of this difference. The angles are then said to be "adjusted." 





STATIONS. 


OBSERVED ANGLES. 


ADJUSTED ANGLES. 


. 


Prince 


41 47 41-79 
81 13 13-78 
56 59 07-39 


41-19 
13-18 
06-79 




Buck 


Hill 


« 


o 

180 + e 




180 00 02-96 
= 180 00 01-16 


01-16 check. 
3 = 0-60 




1-80 -r 



466 



TRIANGULAR SURVEYING. 



The difference between the sum of the observed angles and 180° 
plus the spherical excess (1"-16) is l ff, 80 3 which will make a correc- 
tion for each angle of 0" -60. Subtracting this from the observed an- 
gles, we get the corrected or adjusted spherical angles as in the table. 

692. Having now the length of one side (or base), and the ad- 
justed values of the three angles of a triangle, the other sides might 
be computed by the principles of spherical trigonometry. This 
would be very laborious, but by the help of Legendre's theorem the 
triangle may be computed as if it were a plane one, and the work 
be greatly shortened. The theorem is as follows : 

Legendre's Theorem. " In any spherical triangle, the sides of 
which are small compared with the radius of the sphere, if each 
of the angles be diminished by one third of the spherical excess, 
the sines of these angles will be proportional to the lengths of the 
opposite sides." 

Example. 



STATIONS. 


SPHERICAL ANGLES. 


PLANE ANGLES AND 
DISTANCES. 


LOGARITHMS. 


Prince 


Buck to Hill. 

41 47 41-19 
81 13 13-18 
56 59 6-79 


m. 

19189-80 

40-80 
12-79 
06-41 

284 I 56-10 
24144-18 


4-2830705 

0-1762239 
9-9948811 
9 9235180 

4-4541755 

4-3828124 


Buck 


Hill 




1-16 

Prince to Hill. 
Prince to Buck. 



One third of the spherical excess is subtracted from the spheri- 
cal angles to reduce them to plane angles, which are placed in the 
third column. Using these plane angles, and the given side, and 
applying the sine proportion, we have : 



Log. a 
Log. sin. B 
Co-log. sin. A 

Lost, h 



To find b. 

= 4-2830705 
= 9-9948811 
= 04762239 



Prince to Hill 



= 4-4541755 
= 28456-10 



To find c. 



Log. a 
Log. sin. 
Co-log. sin. A 

Log. c 

Prince to Buck 



= 4-2830705 
= 9-9235180 
= 0-1762239 

= 4-3828124 

= 24144-18 



SPHERICAL SURVEYING, OR GEODESY. 467 

The logarithms of the sides and of the sines of the plane angles 
are placed in the last column. For convenience in calculation, the 
co-log. of angle opposite the given side is taken. 

693. In this manner, starting from the base A B (Fig. 517), a 
single chain of spherical triangles may be computed. If another 
base were measured at E F, a check of the accuracy of the work 
would be afforded by comparing the computed and measured 
values of E F. In the Lake-Survey triangulation of Lake Erie, 
the measured value of the Sandusky base differed from the value 
computed from the Buffalo base through a chain of thirty-six trian- 
gles intervening, by about one inch and a half. 

694. Adjustment of a Triangulation. We have considered the 
measurement and computation of a single chain of triangles pro- 
ceeding from a single measured base A B. Suppose now that the 
observer while at station B had sighted over the line B D, measur- 
ing the angles A B D, CBD, and while at D had measured the 
angles ADB, C D B. We should then have been able to compute 
C D from A B, by using any one of the pairs of triangles ABO, 
BCD:ABC, ACD :ABD, BOD:ABD, ACD. A contra- 
diction is to be expected, as the measurements are not perfect, 
and therefore before beginning the computation of the sides, an 
"adjustment " of the angles must be made, so that their most prob- 
able values alone enter, and no contradiction will appear in the 
computed lengths. 

The question becomes more complicated when bases are meas- 
ured at intervals. Thus, suppose the triangulation adjusted from 
A B as base and E F computed. Another adjustment is needed to 
harmonize this value with the measured value of E F. 

Still further contradictions arise from the introduction of the 
astronomical determination of the direction of a line (or azimuth), 
which must be adjusted for before the work is ready for mapping. 

Consult " Report of the United States Coast and Geodetic Sur- 
vey," 1854, 1864 ; Wright, "Adjustment of Observations," chaps. 
v to ix. On mapping, see " Report United States Coast and Geo- 
detic Survey," 1880. 



468 



TRIANGULAR SURVEYING. 



695. Co-ordinates of the Points. The polar spherical co-ordi- 
nates of a point with respect to another point are these : the length 
of the arc of the great circle passing through the points, and its 
azimuth, i. e.,the angle it makes with the meridian passiDg through 
one of its points. 

The rectangular spherical co-ordinates of a point have for axes 
the meridian passing through the origin, and a perpendicular to it. 
For short distances these may be regarded as in one plane. For 
greater distances new meridians must be taken — say, not farther 
apart than fifty miles. 

Within that limit the successive triangles may be conceived to 
be turned down into the same plane. 

The astronomical co-ordinates of a point are its latitude and 
longitude. These are determined by practical astronomy. 

See "Report of the United States Coast and Geodetic Survey," 
1866, 1868, 1872, 1876, 1880; Chauvenet's "Astronomy," vol. ii ; 
Brunnow's "Astronomy"; Doolittle's "Astronomy." 

The methods of transformation from one system of co-ordi- 
nates to another are of great importance in practice. Two prob- 
lems of common occurrence are the following : 



Problem. Given the latitude and longitude of A, and the 
azimuth and distance from A to B. Required the latitude and 

longitude of B, and the azimuth from 
B to A. 

"When .the triangle sides do not ex- 
ceed fifteen miles, the geodetic lati- 
tudes, longitudes, and azimuths re- 
quired are computed as follows : 

Let K = distance in metres be- 
tween two stations, the latitude and 
longitude of one of which are known. 

L = latitude of first station. 

M = longitude of first. 

Z = azimuth of second station from 

first, counted from the south around 

AE = latitude. by the west, from 0° to 360°. The 




SPHERICAL SURVEYING, OR GEODESY, 



469 



algebraic signs of the sine and cosine of this angle must be care- 
fully attended to. 

L', M', Z', the same things at second station, or quantities re- 
quired, 

e = the eccentricity. 

R = the radius of curvature of the meridian, in metres. 

N = the radius of curvature of a section perpendicular to the 
meridian, in metres. 

Then we have 



T , T K cos. Z . tan. L 

L = L — ^ — — K 2 sm. Z 



K 2 e 2 sin. 2Lcos. 2 Z 



M' = M + 
= M + 



Rarcl" "" 2RNarcl" 4 R 2 (l-e 2 sm. 2 L)§arc 1" 

L - K B cos. Z - K 2 C sin. 2 Z - K 2 B 2 D cos. 2 Z 
K sin. Z • 



N' cos. L' arc 1" 
A'Ksin. Z 



cos. L' 
Z' = Z + 180 - (M' - M) 



sin. ! (1/ + L) 



cos. J (L/ — L) 

when the quantities B, 0, D, A' may be tabulated for given values 
of the latitude entering. Tables for this purpose will be found 
in "Report of the United States Coast and Geodetic Survey," 
1884. 

Example. Given latitude and longitude of station Victory and 
length and azimuth of line Victory-Oswego, to find latitude and 
longitude of Oswego and azimuth of line Oswego-Victory. 

The computation may be conveniently arranged in the follow- 
ing tabular form : 



z 

Z'-Z 

180° 
Z' 


Victory to Oswego 


o 

196 


39 
3 


1! 

39-23 

48-46 


Oswego to Victory 


180 
16 

I 


43 


27-69 


L 
L'-L 

L' 




43 


/ 

13 
13 


// 

06-82 
30-48 


Victory .... 
Oswego .... 


M 

M'-M 


o 

76 


/ 

36 
5 


// 

22-13 
32-92 


43 


26 


37-30 




M' 


76 


30 


49-21 



470 



TRIANGULAR SURVEYING. 



K 

Cos. Z 
B 


4-4168423 
9-9813739 n 
8 5106052 


K2 

Sin. 2 Z 
C 

M'-M 

Sin. £(L'.+ L) 
Cos. i(L'— L) 

ar. co. 

Z'-Z 

* 


8-83368 
8-91488 
1-37716 


(KB cos 
D 


Z)2 


5-8176 
2-3924 


K B cos. Z 

1st terra. 
2d term. 
3d term. 

- (I/-L) 
i(L' + L) 


2-9088214 n 

— 810-63 
0-13 
0-02 


9-12572 

2-5223459 n 
9-8364593 

o- 




8-2100 


o / // 

— 13 30-48 

43 19 52-06 

6 45 


A' 
K 

Sin. Z 
Cos. L' 

ar. co. 

M'-M 


8-5090305 
4-4168423 
9 -4574399, 
0-1390332 


2-3588052 n 
—228-46' 


2-5223459 r 
— 332".94 



697. Prohlem. Given latitude and longitude of two stations, to 
find the distance between them and the azimuth from each to the 
other. 

This is the inverse problem of the preceding. It is solved by 

dividing 

M'-M = A'Ksin. Z sec. L' 

by the first term for 1/ — L, namely, 

L'-L = BKcos. Z, 



whence 



, _ (M'-M)B T , 

taD - Z = (L'-L)A' C0S ' L ' 



which would give us the azimuth at once if we knew L' — L. "We 
therefore seek to compute the smaller terms for the difference of 
latitude in order to obtain K B cos. Z . by subtracting them from 
the known difference of latitude. 



698. In addition to the authorities already quoted, and which 
give the methods in use in the United States, the following list 
may be of service : " Ordnance Survey of Great Britain " ; " Great 
Trigonometrical Survey of India''; "Die Preussische Landestri- 
angulation " ; Bessel, " Gradmessung in Osfcpreussen " ; Jordan, 
"Handbuch der Yermessungskunde " ; Helmert, " Geodasie " : 
Puissant, "'Geodesic" 



PAET Y. 

MARITIME OR HYDRO GRAPHI- 
CAL SURVEYING. 



INTRODUCTION. 

699. The object of this is to fix the positions of the deep and 
shallow points in harbors, rivers, etc., and thus to discover and re- 
cord the shoals, rocks, channels, and other important features of 
the locality. • 

The relative positions of prominent points on the shore are 
first very precisely determined by " Trigonometrical Surveying," 
Part IV. These form the basis of operations, and afford the means 
of correcting the results obtained by the less accurate methods em- 
ployed for filling in the details. 

In addition to the surveying-instruments already described, the 
sextant is much used in hydrographical surveying. When the sex- 
tant is used for determining the position of a point, the angles 
are measured between three lines, passing from the required point 
to three known points. The required point is thus determined by 
trilinear co-ordinates, or by the fifth method, as explained in Art. 8. 



CHAPTER I. 



THE SEXTANT. 

700. Principle. The angle subtended at the eye by lines passing 
from it to two distant objects, may be measured by so arranging 
two mirrors that one object is looked at directly, and the other 
object is seen by its image, reflected from one mirror to the second, 
and from the second mirror to the eye. If the first mirror be 
moved so that the doubly reflected image of the second object be 
made to cover or coincide with the object seen directly, then is the 

desired angle equal to 
twice the angle which 
the mirrors make with 
each other. 

Proof. In Fig. 
519, let D and E be 
two mirrors, perpen- 
dicular to the plane of 
the paper. Let a ray 
of light from the ob- 
ject A be reflected 
from the mirrors D 
and E to the eye at C, 
and B be the other object, looked at directly. Erect perpendicu- 
lars to the mirrors, and prolong them until they meet at F. Pro- 
long the line A D until it meets the line B E at C. The angle 
D F E is equal to the angle which the two mirrors make with each 
other. 

Since the angle of incidence equals the angle of reflection, ADG 
= G D E, and D E F = F E C, 




V-'F 



THE SEXTANT. 473 

then we have : DCE=ADE-DEC 

DCE = 2(GDE-DEF) 

DCE = 2DFE. 

701. Description of the Sextant (Fig. 520). The frame is usually 
of brass, constructed so as to combine strength with lightness. The 

Fig. 520. 




Sextant. 



handle by which it is held is of wood. The index-arm is movable 
about a pivot in the center of the graduated arc. The index-glass 
is a small mirror, attached to the index-arm at the pivot, so as to 
be perpendicular to the plane of the graduated arc. The horizon- 
glass on the left in the figure is attached perpendicularly to the 
plane of the instrument, and parallel to the index-glass when the 
index is at zero. The lower half of this glass is silvered, to make 
it a reflector, and the upper half is transparent. The telescope is 
attached so as to point toward the horizon-glass. Sets of colored 
glasses are used to moderate the light of the sun, when that body is 
observed. 

The sextant has an arc of one sixth of a circle, and measures 
angles up to 120°, the divisions of the graduated arc being num- 



474 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

bered with twice tlieir real value, so that the true desired angle, 
subtended by the two objects, is read off at once. The arc is usu- 
ally graduated to 10' and read by a vernier to 10". 

702. The box or pocket sextant has the same glasses as the larger 
sextant, inclosed in a circular box, about three inches in diameter. 
The lower part, which answers for a handle when in use, screws off 
and is used for a cover. 

The octant has an arc of one eighth of a circumference, and 
measures angles to 90°. 

703. The Reflecting Circle. This is an instrument constructed 
on the same principle, and used for the same purposes, as the sex- 
tant. In it the graduated arc extends to the whole circumference, 
and more than one vernier may be used by producing the index- 
arm to meet the circumference in one or two more points. 

704. Adjustments of the Sextant. 1. To make the index-glass 
perpendicular to the plane of the arc : 

Bring the index near the center of the arc and place the eye 
near the index-glass, and nearly in the plane of the arc. See if the 
part of the arc reflected in the mirror appears to be a continua- 
tion of the part seen directly. If so, the glass is perpendicular to 
the plane of the arc. If not, adjust it by the screws behind it. 

2. To make the horizon-glass perpendicular to the plane of the 
arc: 

The index-glass having been adjusted, sight to some well-defined 
object, as a star, and if, in moving the index-arm, one image seems 
to separate from or overlap the other, then the horizon-glass is not 
perpendicular to the plane of the arc. It must be made so by the 
screws attached to it. 

Another method of testing the perpendicularity of the horizon- 
glass is as follows : Hold the instrument vertically, and bring the 
direct and reflected images of a smooth portion of the distant hori- 
zon into coincidence. Then turn the instrument until it makes an 
angle with the vertical. If the two images still coincide, the glasses 
are parallel ; and, as the index-glass has been made perpendicular 
to the plane of the arc, the horizon-glass is in adjustment. 



THE SEXTANT. 475 

3. To make the line of collimation of the telescope parallel to the 
plane of the arc : 

The line of collimation of the telescope is an imaginary line, 
passing through the optical center of the object-lens, and a point 
midway between the two parallel wires. These wires are made 
parallel to the plane of the sextant by revolving the tube in which 
they are placed. 

To see whether the line of collimation of the telescope is in ad- 
justment, bring the images of two objects, such as the sun and 
moon, into contact at the wire nearest the instrument, and then, 
by moving the instrument, bring them to the other wire. If the 
contact remains perfect, the line of collimation is parallel to the 
plane of the arc ; if it does not, the adjustment must be made by 
the screws in the collar of the telescope. 

4. To see if the two mirrors are parallel when the index is at 
zero : 

Bring the direct and reflected images of a star into coincidence. 
If the index is at zero, then no correction is necessary ; if not, the 
reading is the " index- error ," and is positive or negative, according 
as the index is to the right or left of the zero. 

The "index-error" may be rectified by moving the horizon- 
glass until the images do coincide when the index is at zero, but it 
is usually merely noted, and used as a correction, being added to 
each reading if the error is positive, or subtracted from each read- 
ing if the error is negative. 

705. How to observe. Hold the instrument so that its plane is 
in the plane of the two objects to be observed, and hold it loosely. 
Look through the eye-hole, or plain tube, or telescope, at the left- 
hand or lower object, by direct vision, through the unsilvered part 
of the horizon-glass. Then move the index-arm till the other object 
is seen in the silvered part of the horizon-glass, and the two are 
brought to apparently coincide. Then the reading of the vernier 
is the angle desired. 

If one object be brighter than the other, look at the former by 

reflection. If the brighter object be to the left or below, hold the 

instrument upside down. 
31 



4:76 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

If the angular distance of the object be more than the range of 
the sextant (about 120°), observe from one of them to some inter- 
mediate object, and thence to the other. 

A good rest for a sextant is an ordinary telescope-clamp, through 
which is passed a stick, one end of which is fitted into a hole made 
in the sextant-handle, and the other end of which is weighted for 
a counterpoise. 

THE PRACTICE. 

706. To set out Perpendiculars. Set the index at 90°. Hold 
the instrument over the given point by a plumb-line, and look along 
the line by direct vision. Send a rod in about the desired direction, 
and when it is seen by reflection to coincide with the point on the 
line looked at directly, it will be in a line perpendicular to the given 
line at the desired point. 

Conversely, to find where a perpendicular from a given point 
would strike a line : 

Set the index at 90°, and walk along the line, looking directly 
at a point on it, until the given point is seen by reflection to coin- 
cide with the point on the line. A plumb-line let fall from the 
eye will give the desired point. 

Fig. 521. 



0- 




707. The Optical Square (Fig. 521). This is a box containing 
two mirrors, fixed at an angle of 45° to each other, and therefore 



TEE PRACTICE. 



477 



giving an angle of 90°, as does the sextant with its glasses fixed at 
that angle. It is used to set out perpendiculars. 

708. To measure a Line, One End being inaccessible. Let A B 
be the required line, and B the inaccessible point. 

At A set off a perpendicular, A C, by Art. 706. Then set the 
index at 45°, and walk backward from A in the line of C A pro- 





Fig. 522. 


B 


^^^^ 




^^ 


>-"7 


/ 





D D' A 

longed, looking by direct vision at 0, until you arrive at some point, 
D, from which B is seen by reflection to coincide with C. Then is 
AD = AB. 

If more convenient, after setting off the right angle, set the 
index at 63° 26', and then proceed as before. The objects will be 
seen to coincide when at some point, D'. Then A D' = | A B. If 

Fig. 523. 
B 




the index be set at 71° 34', then the measured distance will be J A 
B, and so on. 

If the index be set at the complements of the above angles, the 



478 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

distance measured will be, in the first case, twice, and in the sec- 
ond case three times the desired one. 

When the distance A D can not be measured, as in Fig. 523, fix 
D as before. Set the index at 26° 34', and go along the line to E, 
where the objects are seen to coincide with each other ; then is A E 
twice A B, and hence E D = A B. 

709. Otherwise, At A set off an angle, as C A D (AD being a 
prolongation of A B). Then walk along the line A C with the index 

Fig. 524. 




set to half that angle, looking at A directly, and B by reflection, 
till you come to some point, C, at which they coincide. Then is 
CA= AB. 



710. To measure a Line when Both Ends are inaccessible. Let 

AB be the required line. At any point, C, measure the angle 
A C B. Set the sextant to half that angle, and walk back in the 
line B prolonged till at some point, D, A, and B are seen to coin- 
cide, as in last problem ; thus making A C = C D. Do the same 
on A C produced to some point, E. Then is D E = A B. 



THE PRACTICE. 
Fig. 525. 



479 




711. All the methods for overcoming obstacles to measurement, 
determining inaccessible distances, etc. (Part I, Chapter V), with 
the transit or theodolite, can be executed with the sextant. 

712. To measure Heights. Measure the vertical angle between 
the top of the object and a mark at the height of the eye, as with 
a theodolite or transit, and then calculate the height as in Part II, 
Art. 578. 

Otherwise. Set the index at 45°, and walk backward till the 
mark and the top of the object are brought to coincide. Then the 
horizontal distance equals the height. 

So, too, if the index is set at 63° 26', the height equals twice the 
distance, and so on. The ground is supposed to be level. 

Fig. 526. 




When the base is inaccessible : Make C = 45°, and D = 26° 34'. 
Then C D = A B. So, too, if = 26° 34', and D = 18° 26'. 

This may be used when a river flows along the base of a hill 
whose height is desired, or in any other like circumstance. 



480 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

713. To observe Altitudes in an Artificial Horizon. In this 
method we measure the angle subtended at the eye between the 
object and its image reflected from an artificial horizon of mercury, 
molasses, oil, or water. The image of the object in the mercury is 
looked at directly, and the object itself is viewed by reflection. The 
object observed is supposed to be so distant that the rays from it, 

Fig. 527. 



which strike respectively the index- glass and the artificial horizon, 
are parallel ; i. e., S and S', Fig. 527, are the same point. 

Then will the observed angle S E S* be double the required augle 
SEH. 

Demonstration. 

a = a', a' = a", and a" = a'". Hence a" = a. 

SES' = fl+fl'" = 2fl = 2SEH. 

714. When the sun is the object observed, to determine whether 
it is his upper or lower limb whose altitude has been observed, pro- 
ceed thus : 

Having brought two limbs to touch, push the index-arm from 
you. If one image passes over the other, so that the other two 
limbs come together, then you had the lower limb at first. If they 
separate, you had the upper limb. 

In the forenoon, with an inverting telescope, the lower limbs 
are parting, and the upper limbs are approaching ; and vice versa 
in the afternoon. 



THE PRACTICE. 

Fig. 528. 



481 




715. To observe very small altitudes and depressions with the 
artificial horizon : 

Stretch a string over the artificial horizon. Place your head so 
that you see the string cover its image in the mercury. Then the 
eye and string determine a vertical plane. 

Then observe, looking at the string by direct vision, and seeing 
the object by reflection, and you have the angle S E N, in Fig. 528, 
the supplement of the zenith-distance. 

Otherwise. Fix behind the horizon-glass a piece of white paper 
with a small hole in it, and with a black line on it perpendicular 
to the plane of the arc. 

Then look into the mercury, so as to see in it the image of the 
line. Your line of sight is then vertical, and the angle to the ob- 
ject seen by reflection is measured as before. 



716. To measure Slopes with the Sextant and Artificial Hori- 
zon. Let AB be the surface of the ground, and A F a horizontal 

Fig. 529. 




482 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

line. Mark two points equally distant from the eye. Measure, by 
the preceding method, the angles (3 and /?', which C A and C B 
make with the vertical C D. Then will half the difference of these 
angles equal the angle which the slope makes with the horizon. 

Demonstration. Continue the vertical line CD to meet the 
horizontal line in F, and draw C E perpendicular to A B. Then 
the triangles ODE and A D F are similar, being right-angled and 
having the acute angles at D equal. Consequently, the angle 
D C E = D A F, which is the angle of the slope with the horizon. 
But D C E = J (/?' — P), hence J (P' — P) = the angle which the 
slope of the ground makes with the horizon. 

If the points A and B be not equally distant from C, but yet 
far apart, this method will still give a very near approximation, the 
error, which is additive, being \ (a' — a). 

Demonstration. 

D C E = P' + a'— 90°, 
~DCE = - p-a + 90°, 
2DCE = /?'-/? + tf'-a, 

717. Oblique Angles. When the plane of two objects, observed 
by the sextant, is very oblique to the horizon, the observed angle 
will differ much from the horizontal anode which is its horizontal 

o 

projection, and which is the angle needed for platting. The pro- 
jected angle may be larger or smaller than the observed angle. 
This difficulty may be obviated in various ways : 

1. Observe the angular distance of each object from some third 
object, very far to the right or left of both. The difference of 
these angles will be nearly equal to the desired angle. 

2. Note, if possible, some point above or below one of the ob- 
jects, and on the same level with the other, and observe to it and 
the other object. 

3. Suspend two plumb-lines, and place the eye so that these 
lines cover the two objects. Then observe the horizontal angle 
between the plumb-lines. 

4. For perfect precision, observe the oblique angle itself, and 



TEE PRACTICE. 



483 



also the angle of elevation or depression of each of the objects. 
With these data the oblique angle can be reduced to its horizontal 
projection, either by descriptive geometry or more precisely by cal- 
culation, thus : 

Let A H B be the observed angle, and A' H B' the required 
horizontal angle. 

Conceive a vertical H Z, and a spherical surface, of which H, 
the vertex of the angle, is the center. Then will the vertical 

Fig. 530. 




planes, A H A' and B H B', and the oblique plane A H B, cut this 
sphere in arcs of great circles, Z A", Z B", and A" B", thus form- 
ing a spherical triangle, A" Z B", in which A" B" = li measures the 
observed angle ; Z A" = Z measures the zenith-distance of the point 
A ; and Z B" = Z' measures the zenith-distance of the point B. 

These zenith-distances are observed directly, or given by the 
observed -angles of elevation or depression. Then we have the three 
sides of the triangle to find the angle B = A' H B'. 

Calling P the half sum of the three sides, we have : 



2 y sin. Z . sin. Z' 

An approximate correction, when the zenith-distances do not 
differ from 90° by more than 2° or 3°, is this : 

^90° _ ?_t?.y tang, i h . sin. 1" - (~^\ cot. \h . sin. 1". 

The quantities in the parentheses are to be taken in seconds. 
The answer is in seconds, and additive. 



717 1 . The advantages of the sextant over the theodolite are 
these : 



484 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

1. It does not require a fixed support, but can be used while 
the observer is on horseback, or on a surface in motion, as at sea. 

2. It can take simultaneous observations on two moving bodies, 
as the moon and a star. 

It can also do all that the theodolite can. Its only defect is in 
observing oblique angles in some cases. By these properties it 
determines distances, heights, time, latitude, longitude, and true 
meridian, and thus is a portable observatory. 



CHAPTER II. 

TRILINEAR SURVEYING. 

718. Tkilikeae Sueveying is founded on the fifth method 
of determining the position of a point, by measuring the angles be- 
tween three lines conceived to pass from the required point to three 
known points, as illustrated in Art. 8. 

To fix the place of the point from these data is much more 
difficult than in the preceding methods, and is known as the 
" Problem of the three points." It will be here solved geometri- 
cally, instrumentally, and analytically. 

719. Geometrical Solution. Let A, B, and C be the known ob- 
jects observed from S, the angles A S B and B S C being there 

Fig. 531. 




measured. To fix this point, S, on the plat containing A, B, and 
C, draw lines from A and B, making angles with A B each equal 



486 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

to 90° — A S B. The intersection of these lines at will be the 
center of a circle passing through A and B, in the circumference 
of which the point S will be situated.* Describe this circle. Also 
draw lines from B and 0, making angles with B C, each equal to 
90° — B S C. Their intersection, 0', will be the center of a circle 
passing through B and C. The point S will lie somewhere in its 
circumference, and therefore in its intersection with the former 
circumference. The point is thus determined. 

In the figure the observed angles, A S B and B S C, are supposed 
to have been respectively 40° and 60°. The angles set off are there- 
fore 50° and 30°. The central angles are consequently 80° and 
120°, twice the observed angles. 

The dotted lines refer to the checks explained in the latter part 
of this article. 

When one of the angles is obtuse, set off its difference from 90° 
on the opposite side of the line joining the two objects to that on 
which the point of observation lies. 

When the angle A B C is equal, to the supplement of the sum of 
the observed angles, the position of the point will be indeterminate, 
for the two centers obtained will coincide, and the circle described 
from this common center will pass through the three points, and 
any point of the circumference will fulfil] the conditions of the 
problem. 

A third angle, between one of the three points and a fourth 
point, should always be observed, if possible, and used like the oth- 
ers, to serve as a check. 

Many tests of the correctness of the position of the point deter- 
mined may be employed. The simplest one is that the centers 
of the circles, O and O', should lie in the perpendiculars drawn 
through the middle points of the lines A B and B C. 

Another is that the line B S should be bisected perpendicularly 
by the line O O'. 

A third check is obtained by drawing at A and C perpendicu- 
lars to A B and C B, and producing them to meet B O and BO', 

* For the arc A B measures the angle A O B at the center, which angle = 180° 
— 2 (90° — A S B) = 2 A S B. Therefore, any angle inscribed in the circumference 
and measured by the same arc is equal to A S B. 



TRILINEAR SURVEYING. 487 

produced, in D and E. The line D E should pass through S ; for, 
the angles BSD and BSE being right angles, the lines D S and 
S E form one straight line. 

The figure shows these three checks by its dotted lines. 

720. Instrumental Solution. The preceding process is tedious 
where many stations are to be determined. They can be more 
readily found by an instrument called a Station-pointer, or Choro- 
graph. It consists of three arms, or straight-edges, turning about 
a common center, and capable of being set so as to make with each 
other any angles desired. This is effected by means of graduated 
arcs carried on their ends, or by taking off with their points (as 
with a pair of dividers) the proper distance from a scale of chords 
constructed to a radius of their length. Being thus set so as to 
make the two observed angles, the instrument is laid on a map 
containing the three given points, and is turned about till the three 
edges pass through these points. Then their center is at the place 
of the station, for the three points there subtend on the paper the 
angles observed in the field. 

A simple and useful substitute is a piece of transparent paper, 
or ground glass, on which three lines may be drawn at the proper 
angles and moved about on the paper as before. 

721. Analytical Solution. The distances of the required point 
from each of the known points may be obtained analytically. Let 
AB = c; BC=a; ABC = B; ASB=S; B S C = S'. Also, 
make T = 360° - S - S' - B. Let B AS = U ; B C S = V. Then 
we shall have : 

Cot.U = cot.Tf C -f- S ' T +l), 
\a . sin. S . cos. T / 

V = T - IT, 

n c . sin. U a . sin. V 

b B = —. — ^— ; or, = — = ^— , 

sm. S sin. S 

q a _ ° ' sm - ^ B S Q n __ a . sin. CBS 
sm. S sm. S 

Proof. In the triangle A B S, we have 

Aor> tjao *t»c<t> AB. sin. B A S c . sin. TJ . _ 

sm. A S B : sm. BAS:: AB:SB= = . [1.1 

sin. A S B sin. S L J 



488 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

In the triangle CBS, we have 

• -r. « /-■ • -n^o ™ ^ oo BC. sin. B C S a . sin Y 
sin. B S C : sin. BCS::BO:SB= — - — — — - — = — — — . [2.1 

sm. B S sin. S' 

„ c . sin. U a . sin. Y . „, . _ . ' 

Hence, — : — — - = — : — — — ; whence, c . sin. S . sin. U — a . sm. S . sm. 
sin. 8 sm. S' 

Y = . . [3.] 

In the quadrilateral A B S, we have 
BCS = 360°-ASB -BSC-ABC-BAS; orV = 360° - S - S' 
- B - U. 

Let T = 360° - S - S' — B, and we have V = T — U [4.] 

Substituting this value of V, in equation [3], we get [Trig., Art. 8], 

c . sin. S' sin. TJ — a . sin. S (sin. T . cos. U — cos. T . sin. U) = 0. 

Dividing by sin. U, we get 

/ cos. TJ \ 

c . sin. S' — a . sin. S ( sin. T . . ' ^ — cos. T ) = 0. 
^ sm. U / 

Whence we have 

cos. U . „ c . sin. S' + a . sin. S . cos. T 

= cot. TJ = — — — . 

sin. U a . sin. S . sin. T 

Separating this expression into two parts, and canceling, we get 

.- _ c . sin. S' cos. T 

COt. TJ = . ; — + . 

a . sm. fe . sin. T sm. T 
Separating the second member into factors, we get 

, _ cos. T / c . sin. S' , , \ 

COtIJ= sinTT L.sin.S.cos.T + V ; ° r 

cot.U = cot.Tf C -f S/ + l). 
\a . sin. S . cos. T / 

Having found TJ, equation [4] gives Y; and either [1] or [2] gives SB; 

and S A and SC are then given by the familiar "Sine proportion " [Trig., 

Art. 12]. 

' Attention must be given to the algebraic signs of the trigo- 
nometrical functions. 

Example. A S B = 33° 45' ; B S C = 22° 30' ; A B = 600 feet ; 
B C = 400 feet ; A C = 800 feet. Eequired the distances and di- 
rections of the point S from each of the stations. 

In the triangle ABC, the three sides being known, the angle 
A B is found to be 104° 28' 39". The formula then gives the 
angle B A S = U = 105° 8' 10" ; whence B C S is found to be 94° 8' 
11" ; and S B = 1042-51 ; S A = 710193 ; and S C = 934-291. 



CHAPTER III. 

SURVEYING THE SHORE-LINE. 

722. The High-water Line. The principal points on the high- 
water line are determined by triangulating. The sections between 
these points are surveyed with the compass and chain, by running 
a series of straight lines so as to follow, approximately, the shore- 
line, and taking offsets from the straight lines of the survey to the 
bends in the shore-line. The straight lines can be more accurately 
determined by "traversing" with the transit. 

723. The Low-water Line. In " tidal- waters " this is more 
difficult, because low and bare for only a short time. The sur- 
vey is best made with the sextant, observing from prominent 
points to three signals, by the trilinear method, and sketching, 
by the eye, bends of the shore between the stations observed 
from. 

There should be one to observe and one to record. Let 1 and 
2, Fig. 532, be two points on the low- 
water line, whose position it is desired q 
to determine. The observations taken ^ x c % 
will be as follows : 

(1.) A and B . . . 18° 
B and C . . . 20° 
(2.) B and C . , . 15° v V 

C and D ... 45° ' 2 

Wlien the shore is inaccessible, a base-line must be measured on 
the water, and points on the shore fixed by angles from its ends, as 
in Art. 729. 



490 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

724. Measuring a Base on the Water. 1. By sound. Sound 
travels at the rate of 1,090 feet per second, with the temperature 
at 30° Fahr. For higher or lower temperatures, add or subtract 
\\ foot for each degree. If the wind blows with or against the 
movement of the sound, its velocity must' be added or subtracted. 
If it blows obliquely, the correction will be its velocity multiplied 
by the cosine of the angle which the direction of the wind makes 
with the direction of the sound. 

2. By measuring with the sextant the angular height of the 
mast of a vessel, then we have : 

Distance = height of mast -f- tan. of the angle. 



CHAPTER IV. 

SOUNDINGS. 

725. Ik sounding, the object is to determine the contour of the 
bottom of any river, lake, bay, etc., so that a chart of it may be 
drawn, showing the depth of water at all points covered by the sur- 
vey. The heights of the points on the bottom are referred to the 
surface of the water as a " datum-plane," and contour-lines may 
be determined in the manner described in " Topography." 

For the same extent of surface, however, if the same degree of 
accuracy is required, it will be necessary to measure the height of 
more points in sounding than in topographical surveying, as the 
surface between the points, whose heights are measured, can not be 
seen and sketched. 

726. For depths up to eighteen feet a sounding-rod, graduated 
to feet and tenths, may be used. For greater depths, a lead-line 
marked to fathoms and half-fathoms will be necessary. The size 
of the line and the weight of the lead will depend upon the depth 
of the water. A lead weighing ten pounds will be sufficient for 
depths up to twenty fathoms. Before using a lead-line it should 
be thoroughly wet and stretched, and the length of the line should 
be frequently tested. 

727. Before commencing the soundings, stations should be 
erected on all of the principal points on the shore, such as head- 
lands, bights of bays, etc. 

A good station-mark is a post, set in the ground about three 

feet, leaving about one foot above the surface. The flag-pole is 
32 



492 MARITIME OR HYDRO GRAPHICAL SURVEYING. 

placed in an auger-hole made in the top of the post. The flag-pole 
can readily be lifted out, and the transit set over the center of the 
station. The number of the station should be marked on each 
post, and it should be distinguished by the combination of colors 
on the flag, or by the number and arrangement of cross-pieces on 
the staff. 

A permanent "bench-mark" must be established, and the 
height of the water, when the soundings are made, noted and 
recorded. 

Stations on the water are marked by buoys. A buoy may be 
made of a light wood float, in which is a hole for the flag-pole. 
The float is anchored with a stone, or by some other means. 

728. The position of the station-buoys, and of the boat when 
sounding, is determined in various ways. 

729. From the Shore. A point on the water may be determined 
by observing to it with a transit from two stations on the shore, at 
a given signal or fixed time. In Fig. 533, the length of the line 

Fig. 533. 




A B, and the angles which the lines of sight make with it would 
then be known, and its place would be fixed by angular co-ordi- 
nates. Two observers are necessary. 



730. From the Boat with a Compass. Observe from the boat 
with a prismatic compass, or a Burnier's compass, to two signals on 



SOUNDINGS. 



493 



shore. The place of the boat is then determined, and may be fixed 
on the map by drawing, from the two known points, lines having 
the opposite bearings, and their intersection will be the required 
point. This is' rapid and easy, but not precise. 

731. From the Boat with the Sextant. Observe with the sextant 
to three signals on shore, noting the two angles. Two observers, 
or one observer with two sextants, are necessary. This is the tri- 
linear method, given in Chapter II of this part. 

732. Between Stations. Positions of the boat are thus deter- 
mined only at considerable distances apart, and the boat is rowed 

Fig. 534. 




from one of these points to a second one, and soundings taken at 
regular intervals of time between them. 

The distance apart of the soundings depends on the regularity 
of the bottom, the depth of the water, and the object of the sur- 
vey. Care should be taken to leave no spot unexplored. 

For great accuracy, anchor at some point, and determine its 
place as above, and then proceed to another point, paying out a 
line, fastened to the anchor, and sounding at regular distances. 
Cast anchor at the second point, go back to the first, take up the 
anchor, go on to the second, and then proceed as before. 

733. In a river or narrow water, the soundings may be taken in 
zigzag lines, from shore to shore, at equal intervals of time, as in 
Fig. 535. 

Where soundings can be made through the ice, the position ol 




494: MARITIME OR HYDROGRAPHICAL SURVEYING. 

all the points can be determined by any of the methods of survey- 
ing. This is the most 
Fig. 535. accurate method of 

^m sounding. 

734. On the sea- 
coast the soundings 
must all be reduced to 
mean low spring-tides. 

735. Tide-Gauges. Tidal observations consist in recording the 
heights of the water at stated times. In order to determine this, 
tide-gauges are necessary. The simplest form is a stick of timber, 
graduated to feet and inches, or tenths, and either set up in the 
water, or fastened to the face of a dock, or pier, so that the rise of 
the tide may be noted upon it. The zero-point of each gauge is 
taken at or below the lowest tide, and is referred to a permanent 
"■ bench-mark " on the shore. On account of the difficulty of sus- 
taining a timber of considerable height against the force of the 
wind and waves, several successive gauges are sometimes used — the 
bottom mark on each gauge higher up being on a level with the 
top line of the next lower. Sucn an arrangement is required on 
gentle slopes. 

On the sea-coast, where the waves make the reading of the staff 
difficult, the staff may be attached to a float, inclosed in an upright 
tube, pierced with holes. The holes in the tube should be of such 
a size as to allow the water to find the mean height inside, and yet 
reduce the oscillations to very small limits. Permanent tide-gauges 
should be self-registering. For a description of a self-registering 
tide-gauge, see " United States Coast Survey Eeport," 1853. 

736. "Establishment of the Port." Owing to the obstructions 
which the tidal wave meets with from the formation of the sea-bed 
as it approaches the shore, and the character and direction of the 
channels, the time of high water will differ for different ports in 
the same vicing. In order that navigators, entering a port, may 
be able to find the time of high water, a standard tide-time is 



SOUNDINGS. 495 

established — i. e. , the number of hours at which high water occurs 
after the moon's transit over the meridian. This is called the 
" Establishment of the Port." This time varies with the age of 
the moon. When observed on the days of full or change, it is the 
"Vulgar Establishment of the Port." The " Corrected Establish- 
ment of the Port " is the mean of the intervals between the times 
of the transit of the moon and the times of high tide for half a 
month. This is used for finding the time of high water on any 
given day, and tables are constructed, from observations at the 
principal ports, for finding the correction for semi-monthly in- 
equality. 

737. In rivers, a number of tide-gauges are necessary, at mod- 
erate distances apart, especially at the bends, because the tidal lines 
of high and low water are not parallel to one another. 

The soundings are to be reduced by the nearest gauge, or by 
the mean of the two between which they may be taken. 

738. Beacons and Buoys. Beacons are permanent objects, such 
as piles of stones with signals on them, usually on shoals and dan- 
gerous rocks. 

Buoys are floating objects, Such as barrels, or hollow iron 
spheres or cylinders, anchored by a chain, and variously painted, 
to indicate either dangers or channels. 

Those placed by the United States Coast Survey are so colored 
and numbered that, in entering a bay, harbor, or channel, red 
buoys with even numbers shall be passed on the starboard or right 
hand, black buoys with odd numbers on the port hand or left hand, 
and buoys with red and black stripes on either hand. Buoys in 
channel-ways are colored with alternate white and black vertical 
stripes. 



CHAPTER V. 



THE CHART. 



Fig. 536. 



739. Havixg determined the lines of high and low water, the 
position of the channels, rocks, shoals, etc., 
and the soundings, a chart must be made, 
on which all these are laid down in their 
proper places. For scales, see Arts. 43-45. 
The high-water line is platted like the 
bounding lines of a farm. The points de- 
termined in the low-water line, and the 
positions of the boat, determined by the 
method given in Arts. 728-731, are fixed 
on the chart by one of the methods given 
in Arts. 719-721. Contour curves are 
drawn as in land topography (Part III), for 
These may be indicated by dotted lines, as 




the first four fathoms. 



Fig. 537. 




p^ 






TEE CHART. 



497 



in Fig. 536, or they may be shaded with Indian-ink, as in Fig. 
537. 

Beyond four fathoms, the depths are noted in fathoms and vul- 
gar fractions. 



740. Various conventional signs are used ; some of the princi- 
pal ones are given in Figs. 538-558. 



Fig. 538. 



Fig. 539. 



Fig. 540. 




Rocky shore. 
Fig. 541. 




Rocks always bare. Low, swampy shore. 

Fig. 542. Fig. 543. 



"%'■/:: 




Fig. 545. 



SAND ISLAND 



Rocks some- 
times bare. 

Fig. 544. 

Reef of rocks. 
Fig. 546. 




Sandy shore, with hillocks. 



x 



%SAND sometimes bare ,*f 






:■' 



Fig. 548. 



Fig. 54V. 

(SAND ALWAYS CO VERD 
Fig. 549. 



FISH WEIRS 



— 



DIRECT/ON OF T«E CI 



Fig. 550. 



Anchorage for 
ships. 



Fig. 551. Fig. 552. Fig. 553. Fig. 554. 

n ♦ j, t 

Buoys. Light-house. Anchorage for Wrecks. 



coasters. 



Fig. 555. 



Fig. 556. Fig. 557. 

^ .-.■£' t& 

Signal-house. Rocks always covered. Harbors. 



Fig. 558. 
Channel-marks. 



PAET VI. 

UNDERGROUND OR MINING 
SURVEYING. 



741. It has three objects : 

1. To determine the directions and extent of the present work- 
ings of a mine. 

2. To find a point on the surface of the ground from which to 
sink a shaft, to meet a desired spot of the underground workings. 

3. To direct the underground workings to meet a shaft or any 
other desired point. 

It attains these objects by a combination of surveying and lev- 
eling. 



CHAPTER I. 

SUEVETIXG AND LEYELIXG OLD LIXES. 

742. First Object. To determine the direction and extent of 
the present workings of a mine. 
We have to measure : 

1. Azimuths, or directions right and left. 

2. Lengths or distances. 

3. Heights, or distances up and down, either by perpendicular 
or angular leveling ; usually the latter. 

This being done, the relative positions of all the points are 
known by their three rectangular co-ordinates. 

They are referred, first, to a vertical plane (which may be 
either north and south, or pass through the first line of the sur- 



SURVEYING AND LEVELING OLD LINES. 499 

vey) ; second, to another vertical plane, perpendicular to the pre- 
ceding one ; and, third, to a horizontal datum-plane. 

743. In making an underground survey, the same rules and 
principles apply as to work on the surface. Some differences in 
methods and detail are necessary, on account of the entire depend- 
ence upon artificial light, and the circumscribed limits within 
which the surveyor is obliged to work. 

As the headings and air-ways of a mine are generally driven far 
in advance of the other workings, it is essential that they should be 
surveyed with great accuracy, in order to give an intelligent idea 
of the territory about to be mined. It is also essential, in order 
that they may serve as a base from which to continue and check 
the surveys of the interior portions of the , mine. 

744. Stations. The work may often be much simplified by a 
careful selection of the stations. See that the average distance be- 
tween them is as long as possible ; that they are convenient for 
future use ; and are so chosen that the instrument can be easily 
set over them. It is also important to locate them where they can 
be easily and permanently marked. Frequently a station may be 
so chosen that several different sights can be taken from it — thus 
economizing much time. 

745. Marking the Stations. Whenever possible, all stations 
should be plainly marked with white paint, and given some dis- 
tinguishing number or letter. This is necessary for use in extend- 
ing the surveys at some future time, and also to make the map of 
use when wishing to identify some particular locality in the mine. 
The precise point may be indicated by an iron spud like a horse- 
shoe nail, with a hole through the head large enough to take the 
line of a plumb-bob or plummet-lamp. The spud is driven in a 
crack in the roof, or in a wooden plug which is driven in a hole 
that has been previously drilled. The objections to this method 
are, the length of time it takes to get the spuds in the roof, and 
also the difficulty in using them when the roof is high. Another 
objection is that mischievous workmen will drive the spuds up in 



500 



UNDERGROUND OR MINING SURVEYING. 



Fig. 559. 




the plugs out of sight with the ends of their drills. Probably, as 
satisfactory a way as any to mark the point is to drill a shallow 

hole, about one eighth of an inch 
in diameter, in the center of a 
painted -}-, or a circle about six 
inches in diameter. Fig. 559 
shows a very convenient device 
for marking the stations, and 
plumbing down from them when 
the roof is high. It is made of 
light gas-pipe, about half an 
inch in diameter, and of any 
convenient length. At one end 
is a drill ; the other end is bent 
about three inches out of line, 
and tapered at the end to fit 
into the hole made with the drill. There is also a notch in the end 
large enough to hold the line of a plumb-bob. Attached to the pipe 
are two rings with shanks about an inch in length. The lower one 
is fixed, the other is adjustable with a clamp-screw. The upper 
ring is split in the back wide enough to take a plumb-line easily. 
To use this device in marking the ■ stations, first strike the drill 
against the roof, then twist it around a few times. This will gen- 
erally make a mark large enough to be easily identified. Then 
reverse the instrument, put the handle of the paint-brush in 
the upper ring, adjust to the proper height, and clamp it fast. 
Put the claw, or notch, in the drill-hole and describe a circle, 
and also paint the number or letter. To plumb down from the 
point in the roof, remove the brush, put the plumb-line in the 
small notch, and through the upper ring, which can be easily 
done through the split. Hold the claw with the plumb-line 
in it against the roof at the proper point, then pay out the 
plumb-line until the plumb-bob reaches the bottom, when the 
point can be fixed. When not in use, bring the two rings to- 
gether, gripping the plumb-bob between them, and clamp fast. 
Wrap the cord around the shanks of the rings, and fasten with a 
half-hitch. 



SURVEYING AND LEVELING OLD LINES. 



501 



746. Points for setting the Transit over. These may be made 
in a variety of ways, as a nail in a tie, a chalk X on a rail or stone, 
a X scratched with a measuring-pin, a speck of paint, or a spot of 
white paint with a speck of coal in the center. If the chalked 
X is too coarse, rub away a portion of it with the finger. Spe- 
cial cases may arise where it would be advisable to carry along 
weights of lead with a short piece of brass wire projecting above 
the surface, to give a precise point. A center-mark on the top 
of the telescope will afford the means of placing the transit in 
position under a plumb-bob suspended from the roof. 



Fig. 560. 



747. Giving the Sights. A measuring- pin, if held plumb, with 
a lamp in front, and a little to one side, makes a very good sight. 
The pin should be whitened with chalk to make a background for 
the cross-hair. The cord of a plumb-bob can be 
seen distinctly up to three or four hundred feet, if 
a piece of white paper is held behind it and a light 
is held in front, Care must be taken not to mis- 
take the shadow of the line for the line itself. It is 
difficult to hold the plumb-bob steady unless it can 
-be hung in the iron spuds mentioned in Art. 745, 
or the device shown in Fig. 559 is used. Where 
the mine is smoky, or the sights are very long, sight 
to the center of the blaze of the lamp, which must 
be carefully plumbed over the point. To meet 
cases of this kind, the plummet-lamp has been de- 
vised (Fig. 560). It consists of a brass lamp hung 
in gimbals and supported by two chains. The lamp 
terminates below in a conical plummet. A shield 
at the top prevents the flame from burning the 
string. The sight is taken to the center of the 
flame. These lamps are generally used in pairs, 
for back-and-forward sights. They are inconven- 
ient to use, as they require the iron spuds with a 
hole through the head to support them from the 
top. Where the roof is high, it is difficult to get up 
to the station to put the string through the hole. 





502 



UXDERGROUXD OR MIS IX G SUEYEYIXG. 



If care is taken not to make them too heavy, they can be sup- 
ported with the device mentioned in Art. 745. Another objec- 
tion is the additional load they impose upon the party to carry. 



Fig. 561. 



748. The Transit. The essential features of a transit to be used 
for surveys in mines are that the verniers should be so placed as to 

be easily read by lamp-light, 
and that the marking should 
be very distinct, on account 
of the imperfect light avail- 
able. Again, the instru- 
ment should not be too 
heavy, as there is often diffi- 
cult climbing to be done 
over fallen rock and other 
mine debris. If the instru- 
ment be easily detached from 
its tripod, it will often be 
found a convenience, as 
thereby the load may be 
lightened and the instrument 
itself more carefully carried 
and more fully protected. 
Graduations on solid silver are apt to be tarnished by the pow- 
der-smoke of the mines. 
Some makers claim to ob- 
viate this by making the 
graduations on platinum. 

If the telescope has a 
level attached, see that the 
lamp is not held under it 
for any length of time, as 
the heat may explode it. 
Accidents of this kind have 
occurred, producing serious 
results. 

In one form of mining 





Z- 



SURVEYING AND LEVELING OLD LINES. 503 

transit an extra telescope is attached on one side, as shown in Fig. 
561, and is balanced by a weight on the opposite side. The ad- 
vantage of this form is, that sights may be taken vertically up or 
down, as is sometimes necessary in connecting the underground 
surveys with those on the surface. 

In another form, the extra telescope is attached to the transit- 
telescope, as shown in Fig. 562. 

The diagonal prism, shown in Fig. 211, may be used with ad- 
vantage on the extra telescope. 

749. Taking the Sights. The beginner will at first have some 
trouble in catching the light through the telescope. A little prac- 
tice will overcome this. Hold a lamp a little above the instrument, 
sight over the top of the telescope, and turn it until it points to 
the light which it is desired to observe. Now sight through the 
telescope, and turn it a little each way, until the eye catches the 
light. Clamp the instrument, and move the object-glass until the 
light looks like a large round blur. This will form a background 
on which the cross-hairs can be plainly seen. m Bisect " the blur, 
then focus the object-glass, and the cross-hairs will be so near the 
right place that there will be no trouble to find them in bisecting a 
plumb-line, or whatever else is sighted to. Some instruments have 
a reflector for illuminating the cross-hairs by throwing a light into 
the telescope (Fig. 210). The same result can be accomplished by 
holding a lamp two or three feet in front of the object-glass, and a 
little to one side, so as to be out of the line of sight. 

750. Measuring the Angles. Proceed as in making a traverse 
on the surface, noting whether the angles are to the right or left. 
It is generally more satisfactory to put the vernier at zero every 
time rather than to survey or traverse by the back-angle. The in- 
strument gets some hard usage, and when the surveyor reviews the 
angle, after having moved to the next station preparatory to meas- 
uring a new angle, he has the unsatisfied feeling of not knowing 
whether the upper motion has slipped, or that he read the angle 
wrong before. It is also more troublesome to set the vernier at 
odd degrees and minutes than at 0, in case there should be a slip of 



504 



UNDERGROUND OR MINING SURVEYING. 





BACK-SIGHTS. 


ANGLES. 


FORE-SIGHTS. 


Thus, 


S. 30° 00' W. 


165° 00' L 


N. 45° 00' E. 


as 


S. 30° 00' W. 


15° 00' R 


K 45° 00' E. 



the upper motion. The surveyor should never omit to check the 
reading of his angles, either by noting whether the sum of the two 
readings on each side of the of the vernier is equal to 180° or 
by repeating the angle. The latter method is the most satisfactory. 
If the graduated circle has a double row of figures reading 180° 
each way, and the deflection should be greater than 90°, it is only 
necessary to read the supplement or smaller angle, noting at the 
same time whether it reads to the right or left on the limb. 

The needle-readings, which should always be taken, will pre- 
vent the gross error of getting into the wrong quadrant. 



is the same 
the needle, 



showing that the last course should be N. E. instead of S. W., as 
the angle would seem to indicate. 

The advantage of this method is that it is a little more con- 
venient to use in working out the courses. It also relieves the 
surveyor of the inquiry as to whether his vernier has passed the 
90°, and he should use the larger or smaller angle. He reads the 
vernier as it stands, and lets the needle determine the quadrant. 
It is almost impossible to set up an instrument so solidly that when 
the cross-hairs are put on a given point they will remain there for 
any length of time. For this reason it is best not to begin to 
measure the angle until everything is all ready ; then measure and 
check by doubling it as quickly as can be done with accuracy. 
Occasions sometimes arise in which a surveyor has but a few hours 
in which to make an extended survey. For a necessity of this kind 
the use of three transits will be found to expedite the work very 
greatly. This prevents loss of time in setting the instrument over 
a given point, the work being carried on from the plumb-line of 
one instrument to that of the next. 

751. Plumbing the Shaft. In order that the lines underground 
may be worked from the same meridian as those on the surface, 



SURVEYING AND LEVELING OLD LINES. 505 

they must be deflected from some line whose azimuth is known. 
Should it not be considered justifiable to depend upon the needle 
to determine the azimuth, and should it be impossible to enter the 
mine by a slope or a tunnel, the surveyor will be obliged to resort 
to plumbing the shaft. Two plumb-lines are carefully put into 
some known line on the surface, and their direction, which will be 
in the same line, is again taken at the foot of the shaft, as a me- 
ridian from which all the lines underground are deflected. As the 
two plumb-lines are necessarily but a few feet apart, and as the 
integrity of all the subsequent work depends upon the accuracy 
with which the direction of the line on the surface is reproduced 
by the plumb-lines at the foot of the shaft, it is necessary that ex- 
treme care should be exercised in doing the work. Much time will 
be saved by studying the local conditions of the shaft, and making 
thorough preparations before beginning the work. In the selection 
of wires, iron and steel are excellent, when new, as their strength 
enables a fine wire to support a heavy weight. The objection is 
that they rust and become treacherous, breaking at most inoppor- 
tune times. Hard-rolled brass wire, though free from this objec- 
tion, has to be very carefully used, as it is liable to kink, and then 
break. If it slips out of the hands while attaching the weights at 
the bottom, it will fly up the shaft in an almost inextricable tangle. 
Copper stretches and the weights have to be carefully watched to 
see that they do not touch the bottom of the vessel in which they 
are suspended. On the whole, however, it seems to give the best 
satisfaction. Have the wire wound on two strong reels, set in 
frames which can be securely anchored. The reels should have 
stops, so that the weights can be held at any point that may be 
desired. 

752. Suspending the Wires. Nail two boards on the sides of 
the head-frame, at right angles to the line of sight, and about four 
feet from the ground. Place on each of these boards a scantling 
about twelve feet long, letting one end rest on the ground a little 
out of the line of sight. The "upper end should project over the 
shaft far enough to clear the sides. Put the reels in position, 
about twenty feet back from the shaft, and also a little out of the 



506 UNDERGROUND OR MINING SURVEYING. 

line of sight, and anchor them securely. Fasten weights of about 
five pounds each to the ends of the wires, and pass them over the 
ends of the scantlings. Then pay out the wires until the bottom 
of the shaft is reached. Bring the wires approximately into line 
by tapping the scantlings with a hammer. In the mean time the 
assistants at the foot of the shaft will attach the large weights and 
place them in pails of water. When the signal is given that all is 
right below, the wires are brought precisely into line, putting in 
the wire farthest from the instrument first, then bringing the other 
to it. This can be very easily and accurately done by tapping the 
scantling gently with a hammer. Examine the wires from the top 
to- the bottom of the shaft to be sure they touch no projecting 
points. Make all secure at the surface, and, before taking up the 
instrument to go below, review the work, to be sure that all is cor- 
rect. Be very careful that no work is done over the head of the 
shaft while men are at work in the shaft at the foot, lest accidents 
should occur. At the bottom of the shaft nail two boards across 
the foot-frame, the same as at the surface. On these place two 
other boards, about ten inches wide and one quarter of an inch 
apart, and reaching across the shaft so that the wires will swing 
freely in the crack between them. These boards serve as a rest for 
the hand in steadying the vibrations of the wires. They also pre- 
vent drops of water from falling into the pails and producing cur- 
rents which will move the weights. Take a small piece of board 
and bevel one edge slightly with a knife. Then lay it across the 
crack between the boards, and bring the beveled edge slowly up to 
one of the wires until it almost touches. Make a mark on the 
edge where it bisects the wire, then watch to see if the wire is per- 
fectly still. In deep shafts the oscillations of the wire are very 
slow, and it is trying to the eye to watch them through the tele- 
scope until they are perfectly still. 

Sometimes wires may be steadied by uniting them with a 
thread or string slightly shorter than the distance between 
them. The weights are also sometimes placed in oil or mer- 
cury. Molasses has also been suggested. If it is impossible to 
perfectly steady the wires, fasten them at the mean of the oscil- 
lations. 



SURVEYING AND LEVELING OLD LINES. 507 

753. Getting the instrument into line is not an easy task for 
the beginner, owing to the difficulty in distinguishing between the 
lines when looking through the telescope. This is overcome by an 
assistant holding a white paper with a light alternately in front of 
and behind the wire farthest away. Another method is to put a 
couple of round rings in the first wire, and then the second wire 
can be seen through the openings in the rings. Another very good 
way is to tack a piece of sheet-iron, of about eight by ten inches, 
to a piece of board of the same size. Make a hole about one six- 
teenth of an inch in diameter in the center of the sheet-iron, and 
at the height of the center of the blaze of a mine-lamp above the 
board. Bend the sheet-iron so that it will be slightly convex with 
the bend at the hole. Place this contrivance behind and as close 
as possible to the rear wire, with the small hole bisecting it. 
Place a lighted lamp behind the sheet-iron so that the blaze will 
cover the hole. Put a small piece of board with white paper 
tacked on it behind the first wire ; also a lighted lamp in front. 
The instrument can now very quickly be brought into line with 
the first wire, and the point of light at the second. Verify by 
holding white paper, with a light, behind the second wire, and 
noting whether it is entirely concealed by the other wire. 

If possible, use two transits, placed on opposite sides of the 
shaft, then verify by seeing if they bisect each other's plumb-lines. 
Do not try to set up the instrument too far away, as it increases 
the difficulty of getting a clear sight of the wires. Watch, also, 
that the shadow of the wire is not mistaken for the wire itself. 
When all is completed, mark the line permanently for future use. 
Where great accuracy is required, plumb the shaft several times, 
and take the mean, depending also upon which of the several 
plumbings has been done with the least probability of error. 

754. Second Method. When there are two shafts convenient to 

each other, let a plumb-line down each shaft ; then connect them by 

a careful survey, both on the surface and underground. Calculate 

the course between the lines on the surface. Calculate also the 

course between the wires underground from an assumed meridian. 

The difference between the two courses will be the correction to be 
33 



508 UNDERGROUND OR MINING SURVEYING. 

applied to the underground courses to make them correspond with 
the azimuth assumed on the surface. 

755. Third Method. Use a transit with a telescope outside the 
standards (Fig. 561). Place the instrument in line directly over 
the shaft, then produce the line to the foot of the shaft by revolv- 
ing the telescope so as to sight directly down the shaft. Get two 
points as far apart as possible at the foot of the shaft, then stretch 
a fine wire carefully over them, producing the line far enough to 
make a convenient station over which the transit can be set. In 
shallow shafts, where communication between the top and bottom 
is easy, the wire may be lined in directly with the instrument. 

756. Fourth Method. If no local attraction exists, and extreme 
accuracy is not required, use the needle. The needle can be read 
to within five minutes, and the errors have the 'probability of cor- 
recting each other in the different courses taken. If there is only 
time and means to do ordinary work, it is better to depend exclu- 
sively upon the needle than upon plumbing and deflections poorly 
done. 

The beginner should remember that the greatest care is neces- 
sary, and that, when his best has been done, there are possibilities 
of error. A surveyor who appreciates these errors will not fail to 
verify his work by repetitions at a later date ; as, by making a con- 
nection with other openings to the surface, such as a drill-hole, an 
opening for air, or a connection through a neighboring mine, 
should such an opportunity present itself. 

757. Keeping the Notes. These will depend very much upon 
the character of the work to be done. Some surveyors prefer to 
use two note-books. In one are recorded all the instrumental work 
done with the transit, together with the stations, and whatever 
explanatory remarks may be necessary. In another, made es- 
pecially for the purpose, are kept all measurements and refer- 
ences, accompanied with a sketch showing where they were taken. 
Where the party is large enough, it may be divided so that both of 
these kinds of work may be kept going at the same time. Another 



SURVEYING AND LEVELING OLD LINES. 509 

method, much used, is to keep all the work in one book, where 
everything will be all together when it is wanted. By having the 
figures represent certain things when in particular places, and the 
use of a few symbols and small sketches in special localities, a 
note-book kept in this manner can generally be made to convey all 
needed information. Below will be found the right- and left-hand 
pages of a note-book kept in this manner ; also a map showing the 
portion of the mine included in the survey of which the notes are 
a part. 

In the first column are the numbers of the stations ; also P 
X, indicating that the station is marked, and in what manner. 
In the second column are the needle-courses of the back-sights. 
The third column shows the angles, with E. and L. for right and 
left. Fourth column, the needle-courses of the fore-sights ; the 
corrected courses can afterward be placed above them in red ink. 
Fifth column, distances. Sixth column, slopes, and whether db. 
Seventh column, height to roof. On the right-hand page, station 
1 would be called out by the chairman as follows : Produce 1 and 2 
back. At 12, 4 right ; at 20, pillar 7 right ; at 25, 2 left ; at 50, 
leave point for future reference ; at 0, 5 right and 9 left ; at 25, 3 
right and 8 left ; at 58, 1 right and 10 left ; at 58, entrance right, 
8 wide and walled ; at 100, 9 right and 3 left ; at 119, entrance 
right, 8 wide and walled ; at distance, 8 right and 2 left, etc. 

There will occur to the surveyor, in practice, various symbols 
and abbreviations which he can use to lessen the labor of recording. 



510 



UNDERGROUND OR MINING SURVEYING. 



March 4, 1886. — Near Foot of Shaft 14. 
Set up at point on line of x 52 and x 51, produced 39'G from x 51. B. S. on x 52, 





BACK-SIGHTS. 


ANGLES. 


FORE-SIGHTS. 


DIS- 
TANCES. 


SLOPE ±. 


HEIGHT 

TO 

ROOF. 


P. x 70. 





o ' 

N. 55-30- W. 
N. 56-50- W. 


O 1 

40-15 L. 


O / 


39-2 


o / 

—0-45 


Rail. 

7-42 


P. x 71. 


1 






S. 84-15 W. 
S. 83-OC' \W 


121-0 


+ 2-05 


Bail. 

10-25 


P. x 72. 


2 


S. 85-30- W. 


22-06- R. 


M. 73-89 W. 
N. 72-10- W. 


126/0 


+ 0-55 


Pare. 
9-73 


From 2, 




N. 73-00- W. 1 


84-25- R. 


S. 10-46 W. 
S. 12-. 5- W. 


93-0 


+ 7-35 


Pave. 
5-21 




3 


N. 73-00- W. J 


12-29- L. 


H". 86-' 8 W. 
N. 86-05- W. 


104-3 


+ 1-02 


Pave. 
11-43 


P. x V. 1. 


A. 


N. 86-00- W. ] 


74-27- L. 


S. 19-25 W. 
S. 19-80- W. 


84-5 


+ 10-02 


Tie. 
4-23 


P. x 74. 


4 


N. 86-00- W. J 


89-55- L. 


TS. 3-57- E. 
N. 4 00- E. 


41-8 


-3-01 


Tie. 
6'75 


P. x 75. 


5 


N. 4-00- E. 


88-39- L. 


S. 84-42- E. 
S. 84 40 E. 


78-3 


-0-32 


Tie. 
7-21 


P. x 76. 


6 


S. 84-45- E. 


10- 09- R. 


S. 74-33- E. 
S. 74-25- E. 


1257 


— 0-22 


Eail. 
7-35 


P. x 77. 


7 


S. 74-30- E. 


16-59- L. 


N. 88-28- E. 
N. 89-00- E. 


144-9 


-0-08 


Pave. 
14-12 


From 77. 




S. 86-35- E. ' 


3-07- L. 


N. 85-21- E. 
N. 89-55- E. 


217-0 


—0-15 


Pave. 
7-52 


P. x H. From 


77 


S. 86-35- E. ► 


43-17- R. 


K 48-15- W. 
N. 50-30- W. 


73-6 


-4-12 


Pave. 
6-25 




8 


S. 86-35- E. J 


33-34- R. 


S. 57-58- E. 
S. 53-35- E. 


99-3 


-0-80 


Eock. 

7-15 




9 


S. 53-35- E. 


37 16- L. 


N. 85-46. E. 
N. 89-10- E. 


N. S5 c -47' E. Errr 


r 0°-01'. 






Begin at P. x V. 1 above t 


o run short cha 


mbers. 




P. x V. 2. 


1 


S. 80-45- E. 


Sl-44- R. 




36-40 


-2-01 


Pave. 
4-92 


P. x V. 3. 


2 


S. 1-00- W. 






20-00 


+ 15 OS 


Pave. 
523 


P. x V. 5. 


3 


N. 85 15- E. 


81-24- L 




79-80 


-6-30 


Pave. 
5-21 




4 


S. 3-50- W. 






43-00 


+ 25-00 


Eail. 
8-20 


Set up 


at © 53 on line between x 


74 and x 75. 


B. S. on x 74. 








S. S4-42- E. 
S. 84-40- E. 


80 03 L. 










P.N. 








X. 15-15- E. 


77-20 


-10-12 


8-7 



E. = right. 
L. = left. 



© = mark for future use. 
] = face and stopped. 



O = width of place is put in circle. 
P. = pillar, also = painted. 



SURVEYING AND LEVELING OLD LINES. 



511 



20- P. R. 

7 — 



® ® © 
40 L. 50 R. 95- E. 

=«= © 

30- E. 102' E. 

® ® <|) 

20- E. 50 E. 45. L. 



13 

12 r 4 



face 



© _®_ 
15 T E. 7-8- E. 
Close on 

P. x 40- 



To P. x 50. 
To P. x 49. 



IS, door 



-50 



21- P. E. 

10-5 

I® 



8-11 
© 
f58-E iilFE 



25 
8-11 3 8 



25 

— 2 

15_ 34- P. L. j>0 60 77- F. E. L . 
4-6 8-8 2-— 6-— 5-6 

7- P. E. 40 60 75 93 © 



50 100 (list. 
FI6 9-3 8-2 

SO-2 © 81- P. R_. 100 (list. 
5-6 ' 5 6 4-7" 5^~ 



310 



1010 10 16 10-13 10-12 



+ 10 Lee. Up ch. 



K „ ■ 20 30- ch. L. 50 65- ch. L. 100 
57 ' d00r - 8* ~5=7~ 9~2 —^5— 8^2 
3- P. L. _20 48 83 

153 12-5 1210 10"- 
K)rb. 32 N. R. 

3-4 43 

®_ 25 



® 
ch. L. 1243 P. x 73 



10-3 



7-3 



+ 10 face 



+ 6- F. E. 



V 



50 P. L. dist. 
70 E. 10-6 ' 8-8 9-6 

25 50 65 80_ 85- P . E. 100 dist. 
fl3 W6 12-2 22-4 8-6~ 9 : 7 77 
20 50 75 100 120 disi 138- P. L. 
M WS T&S W2 15-i WS 15-3 

15 25 P. K 50 70_ 100 125 150 170 200 207 dist. 
12^5 6*5 83 10-3 9-3 10-3 10*3 114 97 7*7 10-3 
5 P. L. 50_ 73^ 
"12-1 103 8-7 

10 15 25 door & P. E. & L. 35 
6^ 2-— 2-9 —10 



10 stopped by water. 



56-9 to © 50 Course 1. 



50 75 85 _97 # 
4-8 ' 3-7* 8 ; 8* 8 Tl* 



12- L. 



Close od © 93* 
Sta. 2 top of page 



Close on x D. 1 in 
Evan Jones's cha. 



10- rb. 25- N. R. „ , „ 20 

1« _. 7 - + llfr - E - Sm 



n_ _30_ 40 80 f a cel 
15-12 10-17 8-18 0-2 J 

18 rb. 44 P. x V. 4 73- rb. 



•10 face 23 E." 



8-7 



+ 12 fr. E. 



_10_ 15 25_ 

511 3-3 3-3 



N. R. ch. 40 + 15- fr. 




~i rr 



10 rb. 20 30 50 dist. 
5-4 7-4 7-3 8-3 83 + ' laC6 ' 



F. R. = far rib. 
JS'. E. = near rib. 



blind entrance, 
distance. 



rb. = rib. 

ch. = chamber. 



hdg. = heading-. 
Pave. = pavement. 



512 UNDERGROUND OR MINING SURVEYING. 

Fig. 563. 




SURVEYING AND LEVELING OLD LINES. 513 

758. Tabling the Survey. On pages 514 and 515 will be found 
a form and the tabling of the above field-notes for office use and 
record. It is best to have a specially prepared book already ruled 
to the required form. All the work of tabling can then be done in 
this book. Should there ever be an occasion to review the work, it 
can easily be found. 

The two double columns headed 1 and 2 are for convenience 
in taking down the numbers as they are called oif from Gurden's 
" Traverse Tables," which are to single minutes, and distances to one 
hundred feet. For convenience in description, we will suppose two 
persons, A and B, to be tabling the above survey. A will take the 
sheet or book on which have been recorded the stations, corrected 
courses, distances, and slopes, and call out the angle, which in the 
present case we will suppose to be K 55° 30' W., distance 39-19. B 
finds this in the book of tables, and on the edge of a sheet of blank 
paper checks the heavy line on the center of the page ; also, the 
two minute columns. A then calls out the distance, 39*19, wdiich 
B sets down on his sheet of paper, and then, using his paper as a 
straight-edge, slides it down the page until he comes to 39, taking 
care to keep the check on the center line. He will then call out 
the numbers under the checks for the minute columns, always 
reading the left-hand one first, to A, who will record them as he 
receives them in columns 1 and 2. The same operation is re- 
peated for the 19. A will then call out the next angle, and, while 
B is searching for it, he will add the numbers given, and, if he has 
time, carry the results out to the proper columns of N., S., E., 
and W. A glance at the course, noting whether it is greater or 
less than 45°, will tell him whether the larger number should be 
put in the column of Latitude or Departure. The same operation 
is repeated for all the courses. 

For convenience in plotting and calculations, the latitudes 
and departures should all be referred to a common origin of 
co-ordinates. In this survey the origin is taken at; the west 
plumb-line of the shaft. Station 51 has been found by previous 
work to have latitude north + 112, and departure west 159. In 
like manner, 51 has been found to have a -f- elevation of 187 '70. 
The slopes and distances should be reduced first, then the 



514: 



UNDERGROUND OR MINING SURVEYING. 











SLOPE DISTANCES 


COURSE AND DIS- 






DIS- 


SLOPE 


REDUCED. 


TANCE REDUCED. 


STATIONS. 


COURSE. 


TANCE. 


+ OR — . 














1st. 


2d. 


1st. 


2d. 










38-99 


0-51 


3214 


22-09 




O ' 


39-19 


O ' 


•20 


•00 


16 


11 


70 


1 N. 55-30 W. 


39-20 


-0 45 


3949 


—0-51 


32 30 


22-20 










99-97 


* 2-66 


119-40 


12 02 






120-96 




2099 


•56 


•96 


•09 


71 ' 


S. 84-15 W. 


121-00 


+ 1-31 


120-96 


+3-22 


120-36 
95-96 


1211 
2815 










99-99 


119 


23-99 


7-04 






125-99 




26-00 


•31 


•95 


•28 


72 


N. 73 39 W. 


126-00 


+ 0-41 


125-99 


+ 1-50 


120-90 
99-77 


3547 
6-74 










99-95 


1-80 


3 99 


•27 


73 is 20 beyond 




104-28 




4-30 


"77 


•28 


•02 


station 


N. 86 08 W. 


104-30 


+ 1-02 


104-23 


+ 2^57 


104-04 


7-03 










40-94 


215 


40-90 


2-82 






41-74 




•80 


•04 


•73 


05 


74 


N. 3 57 E. 


41-80 


— 3 01 


41-74 


—219 


41-63 


287 










73-00 


118 


77-67 


7-20 


1 




78-30 




•3J 


•00 


•30 


•03 


75 


S. 84 42 E. 


78-30 


-0 52 


73-30 


—118 


77-97 


7"23 










120-00 


1-57 


115-65 


31-97 






125-70 




5-70 


•07 


5-49 


1-51 


76 


S. 74 33 E. 


125-70 


-0 45 


125-70 


—164 


121-14 


3b -48 






244-30 




14000 


0-82 


139-95 


375 








4-90 


•02 


4-90 


•12 


77 


ST. 88 28 E. 


144-90 


-0 20 


141-90 


—084 


ll^SS 


3-S7 


Point on line be- 




5tf-SS> 




55-99 
•90 


0-93 
•00 


47-47 
•75 


29-70 
•47 


tween 77& 50 


S. 57-58 E. 


5690 
50-00 


— 1 00 


56-S9 


—0-98 


43-22 


30-17 


Close on 70 


S. 84 15 TV. 


50-00 


-0 32 


50-00 


-0-46 


49 75 


5-01 






91-65 




91-20 
•45 


1214 
•06 


89-40 

•64 


17-HO 
•13 


From 72 to V. 5 . 


S. 10 46 TV. 


92-46 


+ 7-35 


91-65 


+ 12 20 


Wol 


1713 


From 20 back of 




&2-&Z 




S2-72 

•49 


1463 
09 


78-28 
•20 


27-59 
•07 


73 to V. 1 . . . . 


S. W 25 TV. 


8450 


+ 10 02 


"S3^1 


+14-72 


7>-45 


27^66 










209-99 


•92 


.209-31 


17-02 


Old eta. 




216-99 




7-00, 


•03 


697 


•57 


From 77 to 40. . . 


N. 85 21 E. 


217-00 


—0 15 


21699 


-0 95 


216-25 


17 59 






73'^ 




72-31 


5-35 


54-46 


48-6.1 








•6S 


•05 


•37 


•33 


From 77 to H. . . 


K 48 15 TV. 73-68 


-4 12 


78-49 


—5 40 


54-83 


4.>94 




i 




99-00 


•S6 


53-93 


52-51 


Old sta. 


i 99S0 




•30 




•25 


16 


From 77 to 50. . . 


S. 57 58 E. 99-30 


—0 30 


9930 


^36 


^41S 


52 67 


From V. 1 to r 


36-38 




35-93 
•40 


1-27 
•01 


35 oS 
•37 


5-79 
•06 


V. 2 


S. 80 45 E. 36-40 


—2 01 


36-33 


— H 15 


35-90 


5-55 


■d 


19-30 








19 00. 

•so 


0-33 

•ill 


V. 3 % 


S. 1 00 TV. ! 20-00 


+ 15-08 


19-30 


+ 5 2-2 


19-30 


0-44 


1- 


79-28 




73-49 
•79 


894 
•09 


75-73 

•25 


6- "4 

- 


V. 5 % 


K 85 15 E. 79-80 


-6 30 


79"i9 


-4H)3 


79-01 


6-56 


43-00 




42*3 
•67 


11-96 

19 






Close to D. 1. 


S. 3 50 W. 44-70 


+ 15 46 


43-00 


+ 1215 


42-90 




From 74 to © 53. 


S. 84-42 E. 53-00 


-0-52 


53-00 


-0-50 


5-2 77 


490 








75-75 


13-63 


72 36 


19-73 




75-97 




•19 


■04 


34 




From 53 to N. 


N. 15-15 E. 77-20 


—10-12 


To 97 


—13-67 


73 30 


19-95 



SURVEYING AND LEVELING OLD LINES. 



515 



N. 


s. 


E. 


w. 


ALGEBRAIC 

SUM OF 
LATITUDES. 


ALGEBRAIC 

SUM OF 

DEPARTURES. 


ALGEBRAIC- 
SUM OF 
SLOPES. 


HEIGHT 

OF 

ROOF. 


22-20 


12-11 




32-30 


+ 112-00 
+ 134-20 

+ 122-09 


—159-00 
— 191-30 

-311-66 


+ 187-07 
+ 186-56 

+ 169-78 


See sta.50. 
Kail. 

7-42 




120-36 


Rail. 
10-25 


35-47 






120-90 


+ 157-56 


—43256 


+ 191-28 


Pave. 

9-73 


7-03 






104-04 


+ 164-59 


-536-60 


+ 193-85 


Pave. 
11-43 


41-63 




2-87 




+ 206-22 


—533-73 


+ 191-66 


Tie. 

6-75 




7-23 


77-97 




+ 198-99 


—455-76 


+ 190-48 


Tie. 

7-21 




33-48 


121-14 




+ 165-51 


-334-62 


+ 188-84 


Pail. 
7-35 


3-87 




144-85 




+ 169-38 


— 189-77 


+ 188-00 


Pave. 
14-12 




30-17 


48-22 




+ 139-21 


— 141-55 


+ 187-02 






5-01 




49-75 


+ 134-20 


-191-30 


+ 186-56 




88-00 


88-00 


'fmW 


395-05 






90-04 




17-13 


+ 67-52 


-449-69 


+ 203-48 


Pave. 
5-21 




78-48 




27-66 


+ 86-11 


—564-26 


+ 208-57 


Rock. 
4-23 


17-59 




216-28 




+ 186-97 


+ 26-51 


+ 187-05 


Pave. 

7-52 


48 83 






48-94 


+ 218-42 


—244-60 


+ 182-60 


Pave. 
6-25 




52-67 


84-18 




+ 11671 


-105-59 


+ 187-14 


Rock. 

7-15 




5-85 


35-90 




+ 80-26 


-528-36 


+ 207-29 


Pave. 
4-92 




19-30 




0-34 


+ 60-96 


—528-70 


+ 212-51 


Pave. 
5-23 


6-56 




79-01 




+ 67-52 


-449-69 


+ 203-48 


Pave. 
521 




42-90 
4-90 


52-77 


2-87 


-J- 24-62 
-201-30 


— 452-56 

— i80-96 


+ 215-63 
+ 190-86 


Rail. 

8-20 
Pave. 

7-50 


73-30 




19-98 




+ 274-60 


—460-98 


+ 177-19 


Rail. 
8-45 



516 UNDERGROUND OR MINING SURVEYING. 

corrected horizontal distances placed over the others in red 
ink. 

Problem. It is desired to drive the heading from H so that it 
will intersect the slope at X. Required^ the course and distance. 
From the columns of total latitudes and departures in the sheet 
of calculations take : 



Latitude. 


Departure. 


N = -f 274*60 


-460-98 


H = + 218 -42 


- 214-60 



+ 56-18 -216-38 

Tangent, of course, equal departure divided by latitude, 
log. 216-38= 2*3352171 
log. 56-18 = 1-7495817 
tern. 75 — 27 = 10-5856354 = T5~ — 27'= course. 



losr. 56-18 = 1-T495S17 



cos. 75° 27' = 9-4000625 



2-3495192 = 223*62 = distance. 
X, being north and west of H, shows the course to be X. vV. . 
or X. 75° -26 W. 223 -36. 

Unless in special cases where great accuracy is required, the 
more common method of solving this and similar problems is to 
take the course and distance from the map with a protractor and 
scale, this being sufficiently accurate for all practical purposes. 

759. Making the Map. If the map is to be much handled, use 
the best quality of cloth-backed paper. The edges should be bound 
with linen tape, which, if sewed, should be double-stitched, with 
about three stitches to the inch. If the stitches are made closer 
than this, the binding will break off in the line of the needle-holes. 
Ascertain from existing maps, or whatever data may be at hand, 
the most advantageous direction for the meridian of the survey to 
assume on the map. Fix also upon a point for the origin of co- 
ordinates. Begin at the origin and rule the paper into five- or ten- 
inch squares, parallel with the meridian of the survey. Very great 
care is required in doing this work, in order to make all the squares 
check precisely with the scale and be rectangular. Owing to the 



SURVEYING AND LEVELING OLD LINES. 517 

expansion and contraction of the paper, the work of laying out the 
squares should be concluded on the same day it is started. In 
addition to the underground workings, the map should show all 
land-lines, dwellings, roads, streams, ponds of water, and any 
other features of the surface that may have a bearing on an intel- 
ligent working of the mine. Both surveys should be referred to 
the same origin of co-ordinates. In plotting an underground trav- 
erse, it is generally more convenient to locate only every fifth or 
tenth station by its co-ordinates, and use a protractor for filling in 
the balance. 

Take a paper protractor, and letter it N. S. E. W., and fix it at 
any convenient place on the paper, so its N. and S. points will cor- 
respond with the meridian of the survey. Fasten with weights ; 
then transfer the courses from the protractor to where they are 
wanted on the map, scaling off the distances as required. The 
stations that have been located by ordinates will check the slight 
errors in the plotting from the protractor. Having plotted all the 
courses, proceed to fill in the interior work from the references 
and sketches shown on the right-hand page of the note-book. 

In inking the map, use only colors that will wash. A diluted 
solution of bichromate of potash mixed with India-ink will prevent 
spreading of the lines when touched with a wet tinting-brush. 

The map should show all the survey-stations, stoppings of en- 
trances, inclination of strata, and elevation of the stations above 
tide or other datum. 

When different "levels" are to be represented, with their con- 
necting shafts, etc., " isometrical projection " has been used, but 
" military or cavalier projection " is best. 



CHAPTER II. 



LOCATING NEW LINES. 



760. Second Object. To determine, on the surface of the 
ground, where to sink a shaft to meet a desired point in the 
underground workings. 

To do this, repeat on the surface of the ground the survey 
made under ifc — i. e., trace on it the courses and distances of the 
galleries, or their equivalents (Art. 764). 

The chief difficulty is to get a starting-point, and to determine 
the direction of the first line. 



761. When the Mine is entered by an Adit (Fig. 564). Set 
the transit at the entrance, and get the direction of the adit, 

j and prolong it up the 

Fig. 564. hill — i. e., in the same 

vertical plane. The 
third adjustment is here 
important. 

If the line has to be 
prolonged by setting the 
instrument farther on, 
the second adjustment 
is important. 




762. When the Mine is entered by a Shaft. Get the magnetic 
bearing of the first underground line, at the bottom of the shaft, 
with great care. Bring up the end of the line through the shaft 
by a plumb-line, and set the compass over this point. Set out a 



LOCATING NEW LINES. 



519 



line with the same bearing and length as the first underground line, 
and repeat the succeeding courses. 

When the compass can not be set over the point, proceed 
thus : 

1. Find, by trial, a spot, as B (Fig. 565), which is in the cor- 
rect course, and measure off a distance equal to the length of the 
first underground course, and 

then proceed as before. Fig, 565. g 

2. Otherwise. Set up an y- ^-^ 
where, as at A' (Fig. 566), take „.--'' 

the bearing and distance of A 
from A' ; run a line correspond- 
ing with the one underground, 
from A' to B'. Repeat the 
course A' A from B' B ; then A B is the desired line, 





Fig. 666. 




*,*?--"""" 


"H 


....-<^ C 





> 



763. To dispense with the Magnetic Needle. First Method. 
Let down two plumb-lines on opposite sides of the shaft, so that 

their lower ends shall be 
very precisely in the under- 
ground line (see Art. 751). 
Second Method. Set, by 
repeated trials, two transits 
on opposite sides of the 
shaft, so that they shall at 
the same time point to one 
another, and each, also, to one of two points in the underground 
line. They will then give the direction of the line above-ground. 
Third Method. If the telescope of the transit be eccentric, as 
in Fig. 561, set the instrument on a platform over the mouth of 
the shaft, so that the line of collimation of the telescope shall be 
in the same vertical plane with two points in the underground 
line, on opposite sides of the shaft. When the instrument is so 
placed that, in turning the telescope, the intersection of the cross- 
hairs strikes the two points in the underground line, the line 
of sight, when directed along the surface, will give the required 
line. 



520 



UNDERGROUND OR MINING SURVEYING. 



764. Having determined the first line, the courses of the un- 
derground survey may be repeated on the surface ; or the bearing 
and length of a single line be calculated, which shall arrive at the 
desired point. 

Let the zigzag line, A B, B C, CD, D Z (Fig. 567), be the 

courses surveyed underground, A being an adit, or at the bottom of 

a shaft, and Z the point to which it is desired to 

sink a shaft. It is required to find the direction 

and length of the straight line A Z. 

When the compass is used, calculate the latitude 
and departure of each of the courses, A B, B C, etc. 
The algebraic sum of their latitudes will be equal to 
A X, and the algebraic sum of their departures will 

XZ 



Fig. 567. 




be equal to X Z. Then is tan. Z A X = 



XA' 



that 



Fig. 568. 



is, the algebraic sum of the departures divided by 

the algebraic sum of the latitudes is equal to the 

tangent of the bearing. The length of the line 

A Z equals the square root of the sum of the squares of A X and 

XZ ; or equals the latitude divided by the 

cosine of the bearing. 

When the transit is used, instead of refer- 
ring all of the lines to the magnetic meridian, 
as in the preceding case, any line of the sur- 
vey may now be taken as the meridian, as in 
"traversing." 

In Fig. 568 all of the courses are referred 
to the first line of the survey. As before, a 
right-angled triangle will be formed. 

Tan. Z A X = ?-| , and the length of A Z = ^ 

A X -J- cos. X A Z. 

Two or more lines may be substituted for the single line in the 
two preceding cases ; the condition being, that the algebraic sums 
of their latitudes and of their departures shall be equal to those of 
the underground survey. 




A X + X Z s 



or 



LOCATING NEW LINES. 



521 



Fig. 569. 



765. Third Object. To direct the workings of a mine to any 
desired point. 

This is the converse of the second object. We repeat under the 
ground the courses run above-ground ; or their equivalents, as in 
Art. 764. 

In Fig. 569, let AB 3 BC,CD,DY, be the present workings of 
a mine, and Z the shaft to which the workings are to be directed. 

Find the latitude and departure of AZ. 
Then the difference between the algebraic sum 
of the latitudes of the underground courses 
already run, and the latitude of A Z, is the lati- 
tude of the required course ; and the difference 
between the algebraic sum of the departures of 
the underground lines, and the departure of 
A Z, is the departure of the required course. 

The length of YZ equals the square root 
of the sum of the squares of its latitude and 
departure. 




766. Problems. Most of the problems which arise in mining- 
surveying can be solved by an a23plication of the familiar principles 
of geometry and trigonometry : 

1. Given the angle which a vein makes with the horizon, and 

the place where it meets the sur- 
Fig. 570. face, to find how deep a shaft at 

D will be required to strike the 
vein : 



A D 




DC = AD. tan. DAC. 
2. Given the depth of the shaft 
D 0, and the " dip " of the vein, to find where it crops out : 
AD = DC. cot. DAC. 
3. Given the depth of a shaft when the vein "crops out," and 
the " dip " of the vein, to find the distance from the bottom of 
the shaft to the vein : 

B C = A B . cot. ACB. 
If the ground makes an angle with the horizon, then the prob- 
lems involve oblique-angled triangles instead of right-angled tri- 



522 UNDERGROUND OR MINING SURVEYING. 

angles, as in the preceding cases. Their solution, however, is quite 
as simple. 

In the more difficult problems, the measurement of lines is 
required, one or both ends of which are inaccessible. (For a full 
investigation of this subject, see Part I, Chapter V.) 



APPENDIX. 



APPENDIX A. 



SYNOPSIS OF PLANE TRIGONOMETRY* 

1. Definition. Plane Trigonometry is that branch of mathematical sci- 
ence which treats of the relations between the sides and angles of plane 
triangles. It teaches how to find any three of these six parts, when the 
other three are given, and one of them, at least, is a side. 

2. Angles and Arcs. The angles of a triangle are measured by the arcs 
described, with any radius, from the angular points as centers, and inter- 
cepted between the legs of the angles. These arcs are measured by com- 
paring them with an entire circumference, described with the same radius. 
Every circumference is regarded as being divided into 360 equal parts, called 
degrees. Each degree is divided into 60 equal parts, called minutes, and each 
minute into 60 seconds. These divisions are indicated by the marks ° ' ". 
Thus 28 degrees, 17 minutes, and 49 seconds, are written 28° 17' 49" Frac- 
tions of a second are best expressed decimally. An arc, including a quarter 
of a circumference and measuring a right angle, is therefore 90°. A semi- 
circumference comprises 180°. It is often represented by 7r, which equals 
3*14159, etc., or 3^ approximately, the radius being 

unity. 

The length of 1° in parts of radius = 0*01745329 ; 
that of V = 0-00029089 ; and that of 1" = 0*00000485. 

The length of the radius of a circle in degrees, or 
360ths of the circumference = 57*29578° = 57° 17' 
24*8"= 3437*747' = 206264-8". t 

An arc may be regarded as generated by a point, 
M, moving from an origin, A, around a circle, in the 
direction of the arrow. The point may thus describe 
arcs of any lengths, such as AM; A B = 90° = \ 
jt; ABC =180° = 7r 

The point may still continue its motion, and generate arcs greater than a 

* For merely solving triangles, only Articles 1, 2, 3, 5, 6, 10, 11, and 12 are needed. 

f The number of seconds in any arc which is given in parts of radius, radius 
being unity, equals the length of the arc so given divided by the length of the arc of 
one second ; or multiplied by the number of seconds in radius. 
34 



Fig. 571. 




52i APPENDIX A. 

circumference, or than two circumferences, or than three ; or even infinite 
in length. 

While the point, M, describes these arcs, the radius, M, indefinitely 
produced, generates corresponding angles. 

If the point, M, should move from the origin, A, in the contrary direction 
to its former movement, the arcs generated by it are regarded as negative, or 
minus ; and so too, of necessity, the angles measured by the arcs. 

Arcs and angles may therefore vary in length from to + ex in one 
direction, and from to — oo in the contrary direction. 

The Complement of an arc is the arc which* would remain after subtract- 
ing the arc from a quarter of the circumference, or from 90°. If the arc be 
more than 90°, its complement is necessarily negative. 

The Supplement of an arc is what would remain after subtracting it from 
half the circumference, or from 180°. If the arc be more than 180°, its sup- 
plement is necessarily negative. 

3. Trigonometrical Lines. The relations of the sides of a triangle to its 
angles are what is required ; but it is more convenient to replace the angles 
by arcs ; and, once more, to replace the arcs by certain straight lines depend- 
ing upon them, and increasing and decreasing with them, or, conversely, in such 
a way that the length of the lines can be found from that of the arcs, and vice 
versa. It is with these lines that the sides of a triangle are compared.* These 
lines are called Trigonometrical Lines, or Circular Functions, because their 
length is a function of that of the circular arcs. The principal trigonometrical 
lines are Sines, Tangents, and Secants. Chords and versed sines are also used. 
The SINE of an arc, AM, is the perpendicular. MP. let fall, from one 
extremity of the arc, upon the diameter which 
Fig. 572. passes through the other extremity. 

The TAXGEXT of an arc, A M, is the dis- 
tance, A T, intercepted, on the tangent drawn 
at one extremity of the arc, between that ex- 
tremity and the prolongation of the radius 
which passes through the other extremity. 

The SECANT of an arc, AM, is the part, 
T, of the prolonged radius, comprised be- 
tween the center and the tangent. 

The sine, tangent, and secant of the com- 
plement of an arc are called the Co-sine, Co-tangent, and Co-seoant of that 
arc. Thus, MQ is the cosine of AM, B S its cotangent, and S its cose- 
cant. The cosine M Q is equal to O P, the part of the radius comprised be- 
tween the center and the foot of the sine. 

The chord of an arc is equal to twice the sine of half that arc. 
The versed-sine of an arc, AM, is the distance, A P, comprised between 
the origin of the arc and the foot of the sine. It is consequently equal to the 
difference between the radius and the sine. 

* For the great value of this indirect mode of comparing the sides and angles of 
triangles, see Comte's "Philosophy of Mathematics" (Harper's, 1S5T), page 225. 




SYNOPSIS OF PLANE TRIGONOMETRY. 



525 



The trigonometrical lines are usually written in an abbreviated form. 
Calling the arc AM = «, we write, 

M P = sin. a. AT= tan. a. O T = sec. a. 

MQ = cos. a. BS = cot. a. S = cosec. a. 

The period after sin., tan., etc., indicating abbreviation, is frequently 
omitted. 

The arcs whose sines, tangents, etc., are equal to a line = a, are written, 
sin. a, or arc (sin. = a) ; 
tan. #, or arc (tan. = a) ; etc. 

4. The Lines as Ratios. The ratios be- 
tween the trigonometrical lines and the radius 
are the same for the same angles, or number 
of degrees in an arc, whatever the length of 
the radius or arc. Consequently, radius being 
unity, these lines may be expressed as simple 
ratios. Thus, in the right-angled triangle 
ABC, we would have 




sin. A 



tan. A = 



sec. A = — — = 



opposite side 
hypotenuse ' 
opposite side 
adjacent side ' 
hypotenuse 
A C adjacent side ' 



B_C 
AB 

BO 
AC 
AB 



cos. A = — — = 



cot. A 



cosec. A 



AB 
AC 
BC 
AB 
BC 



adjacent side 
hypotenuse ' 

adjacent side 

opposite side , 
hypotenuse 

opposite side ' 



When the radius of the arcs which measure the angles is unity, these 
ratios may be used for the lines. If the radius be any other length, the 
results which have been obtained by the above supposition must be modified 
by dividing each of the trigonometrical lines in the result by radius, and thus 
rendering the equations of the results "homogeneous." The same effect 
w T ould be produced by multiplying each term in the expression by such a 
power of radius as would make it contain a number of linear factors equal 

to the greatest number in any term. 
Fig. 574. The radius is usually represented by 

S' H S * or R. 



5. Their Variations in Length. 

As the point M moves around the 
circle, and the arc thus increases, the 
sines, tangents, and secants, starting 
from zero, also increase ; till, when 
the point M has arrived at B, and 
the arc has become 90°, the sine has 
become equal to radius, or unity, and 
the tangent and secant have become 
infinite. The complementary lines 
have decreased, the cosine being equal to radius or unity at starting and be- 
coming zero, and the cotangent and cosecant passing from infinity to zero. 




526 APPENDIX A. 

When the point M has passed the first quadrant at B, and is proceeding toward 
0, the sines, tangents, and secants begin to decrease, till, when the point has 
reached 0, they have the same values as at A. They then begin to increase 
again, and so on. The table on page 527 indicates these variations. 

The sines and tangents of very small arcs may be regarded as sensibly 
proportional to the arcs themselves ; so that for sin. a", we may write a . 
sin. 1" ; and similarly, though less accurately, for sin. a\ we may write a . 
sin. 1'. 

The sines and tangents of very small arcs may similarly be regarded as 
sensibly of the same length as the arcs themselves.* 

a being the length of any arc expressed in parts of radius, the lengths of 

its sine and cosine may be obtained by the following series : 

a 3 a 5 a 1 

sin. a = a \- h, etc. 

2.32.3.4.5 2.S....7 ' 

n a? a 4 a 6 

cos. a = 1 + h , etc. 

2 2.3.4 2 6' 

Let it be required to find cos. 30°, by the above series. 

SO 
30° = — 77 = \ x 3-1416 = -5236. 

180 6 

Substituting this number for a, the series becomes, taking only three terms 
of it, 

1 _ C5236) 2 + (-o236) 4 _ etc> = 1 _ . 137o78 + -003130 = '866052 ; 
2 24 

which is'the correct value of cos. 30° for the first four places of decimals. 

The lengths of the other lines can be obtained from the mutual relations 
given in Art. 7. Some particular values are given below : 

sin. 30° = \ sin. 45° = \ y 2. sin. 60° = \ V 3. 

tan. 30° = £ +'3. tan. 45° = 1. tan. 60° = V 3. 

sec. 30° = f^3. sec. 45° = V 2. sec. 60° = 2. 

6. Their Changes of Sign. Lines measured in contrary directions from 
a common origin usually receive contrary algebraic signs. If, then, all the 
lines in the first quadrant are called positive, their signs will change in some 
of the other quadrants. Thus the sines in the first quadrant being all meas- 
ured upward, when they are measured downward, as they are in the third 
and fourth quadrants, they will be negative. The cosines in the first quad- 
rant are measured from left to right, and when they are measured from right 
to left, as in the second and third quadrants, they will be negative. The 
tangents and secants follow similar rules. 

The variations in length and the changes of sign are all indicated in the 
following table, radius being unity. The terms " increasing " and " de- 
creasing " apply to the lengths of the lines without any reference to their 
signs : 

* Consequently, the note on page 523 may read thus : The number of seconds in 
any very small arc given in parts of radius, radius being unity, is equal to the length 
of the arc so given divided by sin. 1. 



SYNOPSIS OF PLANE TRIGONOMETRY. 



527 



Lengths and 


Signs of the Trigonometrical Lines for Arcs from 0° to 360°. 


Arcs. 


0° 


Between 0° and 90". 


90° 


Between 90° and 180°. 


180° 


Sine 

Tangent 

Secant 

Cosine 

Cotangent... . 
Cosecant .... 





+ 1 
+ 1 

±00 


+ , and increasing, 
+ , and increasing, 
+ , and increasing, 
+ , and decreasing, 
+ , and decreasing, 
+ , and decreasing, 


+ 1 

±00 
±00 




+ 1 


+ , and decreasing, 
— , and decreasing, 
— , and decreasing, 
— , and increasing, 
— , and increasing, 
+ , and increasing, 





—1 
—1 

Too 

± GO 



Arcs. 


180° 


Between 180° and 270°. 


270° 


Between 270° and 360'. 


360° 


Sine 

Tangent 

Secant 

Cosine 

Cotangent . . . 
Cosecant .... 





—1 
—1 

T 00 
± GO 


— , and increasing, 
+ , and increasing, 
— , and increasing, 
— , and decreasing, 
+ , and decreasing, 
— , and decreasing, 


—1 

±00 

T°° 



—1 


— , and decreasing, 
— , and decreasing, 
+ , and decreasing, 
+ , and increasing, 
— , and increasing, 
— , and increasing, 





+ 1 
+ 1 

Too 



From this table, and Fig. 574, we see that an arc and its supplement have 
the same sine ; and that their tangents, secants, cosines, and cotangents are 
of equal length but of contrary signs ; while the cosecants are the same in 
both length and sign. 

We also deduce from the figure the following consequences: 
sin. (a° + 180°) = — sin. a°. cos. (a° + 180°) = - cos. a°. 

tan. (a° + 180°) = tan. a°. cot. (a + 180°) = cot. a . 

sec. (a° + 180°) = — sec. a°. cosec. (a° + 180°) = — cosec. a°. 

cos. ( — a n ) = cos. a°. 
cot. (— a ) = — cot. a . 
cosec. ( — a°) = — cosec. a°. 
An infinite number of arcs have the same trigonometrical lines ; for, an 
arc a, the same arc plus a circumference, the same arc plus two circumfer- 
ences, and so on, would have the same sine, etc. 

" To bring back to the first quadrant " the trigonometrical lines of any 
large arc, proceed thus : Let 1029° be an arc the sine of which is desired. 
Take from it as many times 360° as possible. The remainder will be 309°. 
Then we shall have sin. 309° = sin. (180° — 309°) = sin. — 129° = — sin. 
129° = - sin. (180° - 129°) = - sin. 51°. 



sin. ( — a ) = — sin. a° 
tan. ( — a°) = — tan. a 
sec. ( — a°) = sec. a°. 



7. Their Mutual Relations. Radius being unity, 



sin. a 









COS. 


a° 




sec. 


a° 


1 






COS. 


a° 


tan. a° x 


cot. 


a c 


= 1. 




1 + (tan. 


a°y 


= 


(sec. a 


3\3 



cot. a = 



cos. a 



sin. a 
1 



cosec. a = — 



sm. a 
(sin. a°y + (cos. a ) 2 = 1.* 
1 + (cot. a°f = (cosec. a°). 



* The square, etc., of the sine, etc., of an arc, is often expressed by placing the 
exponent between the abbreviation of the name of the trigonometrical line and the 



528 APPENDIX A. 

Hence, any one of the trigonometrical lines being given, the rest can be 
found from some of these equations. 

8. Two Arcs. Let a and 5 represent any two arcs, a being the greater. 

Then the following formulas apply : 

sin. (a + 5) = sin. a . cos. 5 + cos. a . sin. 5. 

sin. (a — b) = sin. a 

cos. (a + b) = cos. a 

cos. (a — b) = cos. a 

tan. 
tan. (a + b) = 



. cos. b — cos. 


a 


. sin. 


b. 


. cos. b — sin. 


a 


sin. 


0. 


. cos. b + sin. 


a 


sin. 


b. 


a + tan. 5 









tan. {a — b) 



1 — tan. a . tan. b 
tan. a — tan. b 



1 + tan. a . tan. b 

cot. a . cot. 5—1 

cot. (a +o) = - - . 

cot. b + cot. a 

, , , x cot. a . cot. 5+1 

cot. (a — b) = 

cot. b — cot. a 

sin. a . sin. b = | . cos. (a — 5) — |- cos. {a + 5). 

cos. a . cos. 5 = \ . cos. (« + b) + £ cos. (a — b). 

sin. a . cos. 5 = |- . sin. (# + b) + £ sin. (a — b). 

cos. a . sin. b = | . sin. (# + 5) — \ sin. (# — 5). 
sin. a + sin. 5 = 2 sin. \ (a + 5) cos. ^ (a — 5). 
cos. a + cos. 5 = 2 cos. \ (a + 5) cos. | (a — 5). 
sin. a — sin. 5 = 2 sin. \ {a — 5) cos. |- (a + 5). 
cos. 5 — cos. a = 2 sin. £ (a — 5) sin. ^ (a + 5). 
sin. (« + 5) 



tan. a + tan. 5 = 
tan. a — tan. 5 = 
cot. 5 + cot. a = 
cot. 5 — cot. a = 



cos. a . cos. 5 
sin. (a — 5) 

cos. a . cos. 5 
sin. (a + 5) 

sin. a . sin. 5 ' 

sin. (a — 5) 
sin. a . sin. 5 * 



9. Double and Half Arcs. Letting a represent any arc, as before, we 
have the following formulas: 

sin. 2 a = 2 sin. a . cos. a. 

cos. 2 a = (cos. a) 2 — (sin. a) 2 = 2 (cos. #) 2 — 1 = 1 — 2 (sin. «)'-. 
2 tan. a 2 cot. a 2 



tan. 2 a = 



1 — (tan. a) 2 (cot. a) 2 — 1 cot. a — tan. a 



. (cot. ay - 1 

cot. 2 a = = 4- (cot. « — tan. a). 

2 cot. a - v 



number of the degrees in the arc, thus: Sin. 2 a°, tan.- o°, etc. But the notation given 
above places the index as used by Gauss, Delambre, Arbogast, etc., though the first 
two omit the parentheses. 



SOLUTION OF TRIANGLES. 



529 



sin. -J- a = ^ [£ (1 — cos. a) '. 
cos. \ a — V [£ (1 + cos. a) ]. 



sin. a 1 — cos. a (\ — cos. a\ 

tan. |«= = — . — = V ( — ) . 

1 + cos. a sm. a \1 -f cos. a' 

/l + cog. a\ 

Vl — cos. a' ' 



1 + cos. a 
sin. a 



cos. a 



10. Trigonometrical Tables. In the usual tables of the natural trigo- 
nometrical lines, the degrees from 0° to 45° are found at the top of the table, 
and those from 45° to 90° at the bottom; the latter being complements of 
the former. Consequently, the columns which have Sine and Tangent at top 
have Cosine and Cotangent at bottom, since the cosine or cotangent of any 
arc is the same thing as the sine or tangent of its complement. The minutes 
to be added to the degrees are found in the left-hand column, when the num- 
ber of degrees at the top of the page are used, and in the right-hand column 
for the degrees when at the bottom of the page. The lines for arcs interme- 
diate between those in the tables are found by proportion. The lines are 
calculated for a radius equal unity. Hence, the values of the sines and co- 
sines are decimal fractions, though the point is usually omitted. So too are 
the tangents from 0° to 45°, and the cotangents from 90° to 45°. Beyond 
those points they are integers and decimals. 

The calculations, like all others involving large numbers, are shortened 
by the use of logarithms, which substitute addition and subtraction for mul- 
tiplication and division ; but the young student should avoid the frequent 
error of regarding logarithms as a necessary part of trigonometry. 



SOLUTION OF TRIANGLE3 

11. Right-angled Triangles. Let A B O 

be any right-angled triangle. Denote the sides 
opposite the angles by the corresponding small 
letters. Then any one side and one acute an- 
gle, or any two sides being given, the other parts 
can be obtained by one of the following equa- 
tions : A 



Fig. 575. 




GIVEN. 


REQUIRED. 


FORMULAS. 


a, b 


. c, A, B 


c = y (a 2 + 6 2 ) ; tan. A = -^ ; cot. B = -j. 


a, c 


6, A, B 


a _ a 

& = y(c 2 - a 2 ) ; sin. A = - ; cos. B = -. 


a, A 


b, c, B 


b = a . cot. A ; c = -A— ; B = 90° - A. 
sin. A 


5, A 


a, c, B 


a = 6 . tan. A ; c = • B = 90° - A. 
cos. A 


4 A 


a, 6, B 


a = c . sin. A ; b = c cos. A ; B = 90° — A. 




530 APPENDIX A. 

12. Oblique-angled Triangles. Let ABC be any oblique-angled tri- 
angle, the angles and sides being noted as in the figure. Then any three of 

its six parts being given, and one of them 
being a side, the other parts can be ob- 
tained by one of the following methods, 
which are founded on these three theo- 
rems: 

Theoeem I. — In every plane triangle, 
the sines of the angles are to each other as 
the opposite sides. 

Theoeem II. — In every plane triangle, the sum of two sides is to their dif- 
ference as the tangent of half the sum of the angles opposite those sides is to 
the tangent of half their difference. 

Theoeem III. — In every plane triangle, the cosine of any angle is equal to 
a fraction whose numerator is the sum of the squares of the sides adjacent to 
the angle, minus the square of the side opposite to the angle, and whose de- 
nominator is twice the product of the sides adjacent to the angle. 

All the cases for solution which can occur may be reduced to four : 
Case 1. — Given aside and two angles. The third angle is obtained by 
subtracting the sum of the two given angles from 180°. Then either un- 
known side can be obtained by Theorem I. 

Calling the given side a, we have o = a. . * . ; and c = a - — - — . 

sin. A sin. A 

Case 2. — Given two sides and an angle opposite one of them. The angle 
opposite the other given side is found by Theorem I. The third angle is ob- 
tained by subtracting the sum of the other two from 180°. The remaining 
side is then obtained by Theorem I. 

Calling the given sides a and o, and the given angle A, we have sin. B = 

b 
sin. A . — . 
a 

Since an angle and its supplement have the same sine, the resnlt is am- 
biguous ; for the angle B may have either of the two supplementary values 
indicated by the sine, if b > a, and A is an acute angle. 

C = 180° — (A + B). c = sin. C . a A . 

sin. A 

Case 3. — Given two sides and their included angle. Applying Theorem 
II (obtaining the sum of the angles opposite the given sides by subtracting 
the given included angle from 180°), we obtain the difference of the unknown 
angles. Adding this to their sum we obtain the greater angle, and subtract- 
ing it from their sum we get the less. Then Theorem I will give the remain- 
ing side. 

Calling the given sides a and b, and the included angle C, we have 
A + B = 180° - C. Then 

tan. £ (A — B) = tan. \ (A + B) 



a + 6 

sin C. 



£(A + B) + J(A-B) = A. HA + B)-MA-B)=B. c = a 



sin. A 



SOLUTION OF TRIANGLES. 531 

In the first equation cot. |- may be used in the place of tan. \ (A + B). 

Case 4. — Given the three sides. Let s represent half the sum of the three 
sides = |- {a + ft + c). Then any angle, as A, may be obtained from either 
of the following formulas, founded on Theorem 111 : 






sin. A 



cos. A 



[s(s-a)(s-h)(s-c)] 



be 
J» + C 2 _ a? 



2bc 

The first formula should be used when A < 90°, and the second when 
A > 90°. The third should not be used when A is nearly 180° ; nor the 
fourth when A is nearly 90 ; nor the fifth when A is very small. The third 
is the most convenient f when all the angles are required. 



APPEXDIX B. 



Fig. 511. 




TRANS VER SALS. 

Theoeem I. — If a straight line oe drawn so as to cut any two sides of a 
triangle, and the third side prolonged, thus dividing them into six parts {the 
prolonged side and its prolongation being two of the parts), then will the 
product of any three of those parts, whose ex- 
tremities are not contiguous, equal the product 
of the other three parts. 

That is, in Fig. 577, ABC being the triangle, 
and DF the transversal. BE x AD x CF = 
E A x D C x B F. 

To prove this, from B draw B G, parallel to 
C A. From the similar triangles BEG and 
A E D, we have BG: BE:: AD: AE. From 
the similar triangles B F G and C F D, we have 
CD:CF::BG: BF. Multiplying these proportions together, we have 
BGxCD:BExCF::ADxBG:AE xBF. Multiplying extremes 
and means, and suppressing the common factor B G, we have B E x A D x 
OF = EAxDOxBE. 

These six parts are sometimes said to be in involution. 
If the transversal passes entirely outside of the triangle and cnts the pro- 
longations of all three sides, as in Fig. 578, the theorem still holds good. 
The same demonstration applies. without any change.* 

Theoeem II. — Conversely : If three 
points oe tal'en on tiro sides of a triangle, 
and on the third side prolonged, or on 
the prolongations of the three sides, di- 
viding them into six parts, such that the 
product of three non-corisecutive parts 
equals the product of the other three parts. 
then will these three points lie in the same 
straight line. 

This theorem is proved by a reductio 
ad ahsurdum. 

Theoeem III. — If. from the summits 




This theorem may be extended to polygons. 



TRANSVERSALS. 



533 



of a triangle, lines he drawn, to a point situated either within or without 
the triangle, and prolonged to meet the sides of the tri- 
angle, or their prolongations, 



thus 



them into 



Fig. 579. 



six parts, then will the product of any three non-con- 
secutive parts oe equal to the product of the other three 
parts. 

That is, in Fig. 579, or Fig. 580, 

AE xBFxOD = EB xFOxDA. 
For, the triangle A B F, being cut by the transver- 
sal E C, gives the relation (Theorem I). 

AE x BC x FP 




Fig. 5S0. 




E B x FC x PA. 

The triangle A F, being cut by 
the transversal D B, gives 
DC x FB x PA = AD x CB x 
FP. 
Multiplying these equations to- 
gether, and suppressing the common 
factors PA, C B, and F P, we have 
AExBFxCD=EBxFCx 
DA. 

/^-'"' 1> Theoeem IV. — Conversely : If 

E three points are situated on the three 

sides of a triangle, or on their pro- 
longations (either one, or three, of these points oeing on the sides), so that 
they divide these lines in such a way that the product of any three non-con- 
secutive parts equals the product of the other three parts, then will lines drawn 
from these points to the opposite angles meet in the same point. 
This theorem can be demonstrated by a reductio ad absurdum. 



<.-"**" P ^v. 



COEOLLAEIES OF THE PRECEDING THEOEEMS. 

Coeollaet 1. — -The MEDIANS of a triangle (i. e., the lines drawn from 
its summits to the middles of the opposite sides) meet in the same point. 

For, supposing, in Fig. 570, the points D, E, and F to be the middles of 
the sides, the products of the non-consecutive parts will be equal — i. e., 
AExBFxCD = DAxEBxFC; since AE = E B, B F = F C, CD 
= D A. Then Theorem IV applies. 

Cor. 2.— The BISSECTRICES of a triangle (i. e., the lines bisecting its 
angles) meet in the same point. 

For, in Fig. 579, supposing the lines A F, B D, CE to be bissectrices, we 
have (Legendre, IV, 17) : 

BF:FC::AB:AC) (BFxAC = FCxAB, 

OD:DA::BO:BA whence ■]cDxBA = DAxBC, 
AE:EB::CA:CB) (AExCB = EBxCA. 

Multiplying these equations together, and omitting the common factors, 
we have BFxCDxAE = FCxDAxEB. Then Theorem IV applies. 

Cor. 3. — The ALTITUDES of a triangle (i. e., the lines drawn from its 
summits perpendicular to the opposite sides) meet in the same point. 



534 APPENDIX B. 

For, in Fig. 579, supposing the lines AF, BD, and CE to be altitudes, 
we have three pairs of similar triangles, BCD and F C A, C A E and DAB, 
A B F and E B 0, by comparing which we obtain relations from which it is 
easy to deduce BFx CD x AE = EB xFC xDA; and then Theorem 
IY again applies. 

Coe. 4. — If, in Fig. 579, or Pig. 580, the point F be taken in the middle 
o/BO, then will the line E D oe parallel to B C. 

For, since B F = F C, the equation of Theorem III reduces to A E x CD 
= EB x DA; whence AE:EB::AD:DC; consequently E D is parallel 
toBC. 

Coe. 5.— Conversely : IfK D be parallel to B C, then isBF = FC. 
For, since AE : E B : : AD : D C, we have AExDC = EBxAD; 
whence, in the equation of Theorem III, we must have B F = F C. 
Coe. 6. — From the preceding corollary, we derive the following : 

If two sides of a triangle are divided proportion- 
ally, starting from the same summit, as A, and lines 
are drawn from the extremities of the third side to 
the points of division, the intersections of the corre- 
sponding lines will all lie in the same straight line 
joining the summit A, and the middle of the base. 

Coe. 7. — A particular case of the preceding corol- 
lary is this : 

In any trapezoid, the straight line which joins the 
intersection of the diagonals and the point of meeting 
of the non-parallel sides produced, passes through the middle of the two par- 
allel bases. 

Coe. 8. — If the three lines drawn through the corresponding summits of 
two triangles cut each other in the same point, then the three points in ichich 
the corresponding sides, produced if necessary, will meet, are situated in the 
same straight line. • 

This corollary may be otherwise enunciated, thus : 

If two triangles have their summits situated, two and two, on three lines 
which meet in the same point, then, etc. 

This is proved by obtaining by Theorem I three equations, which, being 
multiplied together, and the six common factors canceled, give an equation 
to which Theorem II applies. 

Triangles thus situated are called homologic ; the common point of meet- 
ing of the lines passing through their summits is called the center of homol- 
ogy ; and the one on which the sides meet, the axis of homology. 



HARMONIC DIVISION. 

Definitions. — A straight line, AB, is 

FlG - 582 ' , said to be harmon ieally divided at the points 

J g q C and D, when these points determine tiro 

additive segments, AC, B C, and two sub- 




HARMONIC DIVISION. 



535 



tractive segments, AD, BD, proportional to one another ; so that A : B C 
: : A D : B D. It will be seen that A C must be more than B C, since A D is 
more than B D.* 

This relation may be otherwise expressed, thus : The product of the whole 
line by the middle part equals the product of the extreme parts. 

Reciprocally, the line D is harmonically divided at the points B and A, 
since the preceding proportion may be written DB: C B : : D A : OA. 

The four points, A, B, 0, D, are called harmonics. The points and D 
are called harmonic conjugates. So are the points A and B. 

When a straight line, as A B, is divided harmonically, its half is a mean 
proportional between the distance from the middle of the line to the two 
points, C and D, which divide it harmonically. 

If, from any point, O, lines be drawn so as to divide a line harmonically, 

these lines are called an harmonic pencil. 
Fig. 583. The four lines which compose it, O A, C, 

OB, O D, in the figure, are called its radii, 
and the pairs which pass through the conju- 
gate points are called conjugate radii. 

Theoeem Y. — In any harmonic pencil, a 

line drawn parallel to any one of the radii 

is divided by the three other radii into two 

equal parts. 

Let E F be the line, drawn parallel to O A. Through B draw G H, also 

parallel to O A. We have, 

GB: OA:: BD: AD; and Fig. 584. 

BH: OA::B0: AC. 
But, by hypothesis, AC:BC::AI^: 
BD. 

Hence, the first two proportions re- 
duce to G B = B H ; and, consequent- 
ly, EK = KF. 

The reciprocal is also true — i. e., 
If four lines radiating from a point 
are such that a line drawn parallel 
to one of them is divided into two equal 
parts oy the other three, the four lines form an harmonic pencil. 

Theoeem YI. — If any transversal to an harmonic pencil be drawn, it will 
be divided harmonically. 

Let L M be the transversal. Through K, where L M intersects B, draw 
E F parallel to A. It is bisected at K by the preceding theorem ; and the 





* Three numbers, m, n, p, arranged in decreasing order of size, form an harmonic 
proportion, when the difference of the first and the second is to the difference of the 
second and the third, as the first is to the third. Such are the numbers 6, 4, and 3 ; 
or 6, 3, and 2 ; or 15, 12, and 10 ; etc. So, in Fig. 582, are the lines A D, AB, and 
AC, which thus give B D : C B : : AD : AC ; or AC : CB : : AD : B D. The series 
of fractions, \, £, ^, \, £, etc., is called an harmonic progression, because any con- 
secutive three of its terms form an harmonic proportion. 



536 



APPENDIX B. 



Fig. 585. 



similar triangles, FMK and L M 0, E K IsT and LNO, give the propor- 
tions 

LH:KM:: OL: FK, andLN: NK: : OL: EK; whence, 
since FK = EK,we have LN:NK::LM:KM. 
Ooeollaey. — The two sides of any angle, together with the bissectrices of 
the angle and of its supplement, form an har- 
monic pencil. 

Theoeem VII. — If from the summits of 
any triangle, ABO, through any point, P, 
there be drawn the transversals A D, B E, F, 
and the transversal E D be drawn to meet A B 
prolonged in F', the points F and W will di- 
vide the base A B harmonically. 

This may be otherwise expressed, thus : 

The line, P, which joins the intersection of the diagonals of any quadri- 
lateral, A B DE, with the point of meeting, C, of two opposite sides prolonged, 
cuts the side A B in a point F, which is the harmonic conjugate of the point of 
meeting, F' of the other two sides, E D and AB, prolonged. 
For, by Theorem I, AF'xBD x CE=F'B xDCx 

by Theorem III, AF x BD x CE = FB 
whence AF:FB:: AF' : F'B. 




EA; 
x DC x E A 



and 



THE COMPLETE QUADRILATERAL. 

A Complete Quadrilateral is formed by drawing any four straight lines, 
so that each of them shall cut each of the other three, so as to give six differ- 
ent points of intersection. It is so called 
because in the figure thus formed are 
found three quadrilaterals; viz., in Fig. 
586, ABCD, a common convex, quadri- 
lateral; EAFO, a uni-concace quadri- 
lateral ; and E B A F D, a bi-concave quad- 
rilateral, composed of two opposite trian- 
gles. 

The complete quadrilateral, A E B C 
D F, has three diagonals ; viz., two inte- 
rior, AC, BD; and one exterior, E F. 

Theoeem VIII. — In every complete 
quadeilateeal the middle points of its 
V three diagonals lie in the same straight 

line. 
A E B C D F is the quadrilateral, and L M X the middle points of its three 
diagonals. From A and D draw parallels to B 0, and from B and C draw 
parallels to A D. The triangle EDO being cut by the transversal B F, we 
have (Theorem I), DFxCBxEA = CFxEBxDA. From the equal- 
ity of parallels between parallels, we have CB=E'B', E A = C A'. E B = 
DB', DA = E' A'. Hence, the above equation becomes DFxE'B'x CA' 




THE COMPLETE QUADRILATERAL. 



537 



Fig. 587. 




= CFxDB'xE'A'; therefore, by Theorem II, the points, F, B', A', lie 
in the same straight line. Now, since the diagonals of the parallelogram 
E A' A bisect each other at N, and those of the parallelogram EBB'D 
at M, we have EN:NA'::EM:MB'. Then M N is parallel to F A', and 
we have E N : N" A' : : E L : L F, or E L = L F, so that L is the middle of 
E F, and the same straight line passes through L, M, and N". 

Theorem IX. — In every complete quadrilateral each of the three diago- 
nals is divided harmonically by the two 
others. 

C E B A D F is the complete quadri- 
lateral. The diagonal EF is divided 
harmonically at G and H byDB and 
A C produced ; since AH, D E, and 
FB are three transversals drawn from 
the summits of the triangle A E F 
through the same point C ; and there- 
fore, by Theorem VII, D B G and A C H 
divide EF harmonically. 
So too, in the triangle A B D, C B, A, C D, are the three transversals 
passing through ; and G and K therefore divide the diagonal B D har- 
monically. 

So, too, in the triangle, A B 0, D A, D B, D C are the transversals, and H 
and K the points which divide the diagonal A harmonically. 

Theorem X. — If from a point, A, 
any number of lines be drawn, cutting 
the sides of an angle POQ, the inter- 
sections of the diagonals of the quadri- 
laterals thus formed will all lie in the 
same straight line passing through the 
summit of the angle. 

By the preceding theorem, the diag- 
onal B C' of the complete quadrilateral, 
BAB'C'CO, is divided harmonically 

at D and E. Hence, A, OP, O D, and Q, form an harmonic pencil. So 
do O A, P, O D', and Q. Therefore, the lines O D, O D', coincide. So 
for the other intersections. 

If the point A moves on O A, the line D is not displaced. If, on the 
contrary, O A is displaced, D turns around the point O. Hence, the point 
A is said to be a pole with respect to the line O D, which is itself called the 
polar of the point A. Similarly, D is a pole of A, which is the polar of D. 
O D is likewise the polar of any other point on the line O A ; and this prop- 
erty is necessarily reciprocal for the two conjugate radii A, OD, with re- 
spect to the lines O P, Q, which are also conjugate radii. Hence : in every 
harmonic pencil, each of the radii is a polar with respect to each point of 
its conjugate ; and each point of this latter line is a pole with respect to the 
former. 



Fig, 588. 




ANALYTICAL TABLE OF CONTENTS. 



PART I. 



LAND-SUR VEYING. 



Chapter I. — General Principles and Fundamental Operations. 



ARTICLE PAGE 

1. Surveying defined 1 

2. "When a point is determined 1 

3. First method 2 

4. Second method 2 

5. Third method 3 

6. Fourth method 4 

8. Fifth method 5 

9. Sixth method 5 

10-12. Kinds of surveying 5 

13. Stages of operation 6 

Making the Measurements. 

14. Measurements required 7 

15. Measuring straight lines 7 

16. Gunter's chain 7 

17. Pins 10 

18. How to chain 11 

19. Tallies.. 12 

20. Chaining on slopes 12 

21. Doing up a chain 14 

22. Tape ', 14 

23. Substitutes for chain 15 

24. Eods 15 

25. Approximate methods 15 

26. Perambulator and odometer 15 

Measuring Angles. 

27. Goniometer 16 

28. Chain-angles 17 

Surveying "without Instruments. 

29. Distances by pacing 18 

30. Distances by visual angles 19 

31. Distances by visibility 19 

32. Distances by sound 20 

33. Angles 21 

35 



| ARTICLE page 

Noting- the Measurements. 

35. Making a map 21 

36. Platting 21 

37. Straight lines 22 

38. Arcs 22 

39. Parallels 22 

40. Perpendiculars 23 

41. Angles 23 

42. Drawing to scale 25 

43. Scales 26 

44. Farm-surveys 26 

45. State-surveys 27 

46. Eailroad-surveys 28 

Scales of equal parts 28 



47. 

IS. 
41). 
50. 
51. 
52. 



Vernier scales 

Reduced scale 

Sectoral scale 

Material for scales. 
Scale omitted 



30 
31 
82 
32 
33 



Calculating the Content. 

54. Horizontal measurement 35 

55. Unit of content 36 

57. Chain correction 36 

58. Boundary-hues 36 

Methods of Calculation. 

59. Classification 37 

60. Arithmetically 37 

61. Eectangles 37 

62. Triangles 37 

63. Parallelograms 38 

64. Trapezoids 38 

65. Trapeziums. 38 

'6Q. Geometrically 39 



510 



ANALYTICAL TABLE OF COXTEXTS. 



PAGE AETICLE 



67. Division into triangles 39 

68. Graphical multiplication 40 

69. Division into trapezoids. 41 

70. Division into squares 42 

71. Division into rectangles 42 

72. Addition of widths 43 



73. Lnstrumentallij . 



74. Eeduction to a triangle 43 

75. General rule 45 

76. Examples 46 

77. Special instruments 47 

78. Planimeters ! 48 

79. By weight 48 



43 80. TrigoiiometricaV.ij „ 49 



Chapter II. — Chain-Surveying. 



82. Surveying by diagonals 

Keeping Field- Notes. 

83. By sketch 

84. In columns 

85, 86. Field-books 

86, 87. Surveying by tie-lines 

88. Chain-angles 

90, 91. Inaccessible areas 

92. Surveying by diagonals 

94. Surveyor's cross 

Optical square 

Diagonals and perpendiculars 

Offsets 

Platting offsets . 

100. Calculating content 

Equalizing 

Combination of methods 

Field-books 

104. Inaccessible areas 

105. Obstacles to measurement 

106-120. Problems on perpendiculars. 

121-125. Problems on parallels 

128, 129. Banging with rods 

To prolong a Line. 

By perpendiculars 

By equilateral triangles 

By symmetrical triangles 

By transversals 

By harmonic conjugates 



93. 

95. 

96. 

97. 

93. 

99, 
101. 
102. 
103. 



130, 
131, 
132, 
133 
134 



135. By the complete quadrilateral . 



To interpolate Points on a Line. 

138. Across a valley 89 

139. Over a hill 89 

141. On water 90 

142. Through a wood 91 

143. To an invisible intersection 91 

Obstacles to Measurement. 

A. When Both Ends of the Line are' ac- 

cessible. 

145. By perpendiculars 92 

146. By equilateral triangles 92 

147. By symmetrical triangles 93 

148. By transversals 93 

B. When One End of the Line is accessible. 
149-151. By perpendiculars 93 

152. By parallels 94 

153. By a parallelogram 94 

154. 155. By symmetrical triangles. . . 95 

156. By transversals 95 

157. By harmonic division 96 

158. To an inaccessible line 96 

159. To an inaccessible intersection. . . 97 

C. When Both Ends of the Line are in- 

accessible. 

160. By similar triangles 97 

161. By parallels 97 

162. By a parallelogram 98 

163. 164. By symmetrical triangles 93 



Chapter III.— Compass-Purveying. 



165. Principle . 100 

166, 167. Definitions 100 

168. The needle 101 

169. The sights 102 

170. The divided circle 103 

171. The points 104 

172. Eccentricity 104 

173. Levels 106 

174. Tangent scale 106 

175. Verniers 107 



176. Tripods 107 

177. Jacob's staff 108 

178. The prismatic compass 109 

179. Defects of the compass Ill 

180. Taking bearings 112 

181. Marking of compass-points 113 

1S2. Beading the vernier 114 

183. Practical hints 114 

1S4. To magnetize a needie 116 

185. Back-sights 116 



ANALYTICAL TABLE OF CONTENTS. 



541 



A.BTICLE PAGE 

186. Local attraction 117 

187. Angles of deflection 118 

188. Angles between courses 118 

189. To change bearings 120 

190. Line-surveying 121 

191. Checks by intersecting bearings. . 122 
192-195. Keeping field-notes 122 

196. Canal-maps 123 

197, 198. Farm-surveying 124 

199, 200. Field-notes 125 

201. Tests of accuracy 126 

202. Method of radiation 126 

203. Method of intersection 127 

201. Eunning old lines 127 

Platting the Survey. 

206. Platting bearings 1 28 

207. With a protractor 129 

208. To close a plat 130 

209. Field-platting 131 

210. With a protractor 132 

211. With paper ruled into squares 132 

212. With a paper protractor 132 

213. Drawing-board protractor 133 

214. With a scale of chords 134 

215. With a table of chords 134 

216. With a table of natural sines 135 

217. By latitudes and departures 135 

Copying Plats. 

219. Stretching the paper 136 

220. Copying by tracing 136 

221. By tracing-paper 137 

222. By topography 137 

223. By blue prints 137 

224. By transfer-paper 138 

225. By punctures 138 

226. By intersections 138 

227. By squares 139 

228. Eeducing by squares 1 39 

229. By proportional scale. . . . 139 

230. By pantagraph 140 

231. By camera lucida 140 

232. Enlarging plats 140 

233. Conventional signs 140 

234. Orientation 141 

235. Lettering 141 

236. Borders 141 

237. Joining paper 141 

233. Mounting maps 141 

Latitudes and Departures. 

239. Definitions 142 

240. Calculation of latitudes and de- 

partures 143 



article page 

241. Formulas 144 

242. Traverse- table 145 

243. Application to testing a survey. . . 148 

244. Application to supplying omis- 

sions 149 

245. Balancing a survey 150 

246. Application to platting 151 

Calculating the Content. 

247. Methods 152 

248. Definitions 153 

249. Longitudes 153 

250. Areas 154 

251. A three-sided field 155 

252. A four-sided field 155 

253. General rule 156 

254. To find east or west station 157 

255-257. Examples 157 

258. Mascheroni's theorem 161 

259. New method of calculating areas. 162 

The Declination of the Magnetic Nee- 
dle. 

260. Definitions 164 

261. Direction of the needle 164 

To determine the True Meridian. 

262. By equal shadows of the sun. .... 1 64 

263. By the north star in the meridian. 165 

264. Times of crossing the meridian. . . 167 

265. By the north star at extreme elon- 

gation 168 

266. Observations 170 

267. Azimuths 170 

268. Setting out a meridian 172 

269,270. Determining the declination. 172 

271. Magnetic declination in the United 

States 173 

272. To correct magnetic bearings 174 

273. To survey a line with true bear- 

ings 176 

Variations of Magnetic Declination. 

274. Kinds of variation 176 

275. Irregular variation 176 

276. Diurnal variation 177 

277. Annual variation 177 

278. Secular variation 178 

279. Determination of change by inter- 

polation 179 

280. Determination of change by old 

lines 180 

281. Effect of secular change 180 

282. To run old lines 181 

283. Eemedy for evils of secular change. 184 



542 



ANALYTICAL TABLE OF CONTEXTS. 



Chapter IV. — Transit-Surveying. 



ARTICLE PAGE 

284. The transit. 185 

285. Surveyor's transit 187 

286. Cross-section of transit 187 

287. The telescope... , 188 

28S. The cross-hairs 190 

289. Instrumental parallax..., ...... . 192 

290. Movement of objective and eye- 

piece.. . .. 192 

291. Supports 193 

292. The indexes .... 193 

293. The graduated circle 194 

294. Movements 195 

295. Levels 195 

296. Parallel plates 196 

297. Watch-telescope 197 

298. The compass 197 

299. The reflector 197 

300. The diagonal eye-piece 198 

301. 302. The engineer's transit 198 

303. The theodolite 200 

304. The goniasmometre 200 

Verniers. 

305. Definitions 201 

306. Illustration 201 

307. General rules 202 

308. Eetrograde verniers 203 

309. Illustration 204 

310. Barometer vernier 204 

311. Circle divided into degrees 205 

312. Circle divided to 30' 206 

313. Circle divided to 20' 208 

314. Circle divided to 15' 210 

315. Circle divided to 10' 210 

316. Beading backward 210 

317. Arc of excess 211 

318. Double verniers 211 

319. Compass-vernier 212 



article page 

Adjustments. 

320. Object and necessity 213 

321. Three requirements 213 

322. First adjustment 214 

323. Second adjustment 215 

324. Third adjustment 218 

325. Centering the eye-piece 219 

326. Centering the object-glass 219 

327. Fourth adjustment 221 

328. Fifth adjustment 221 

The Feeld-Work. 

329. To measure an angle 222 

330. Seduction of high and lovr objects 223 

331. Notation of angles 223 

332. To repeat an angle 224 

333. Angles of deflection 225 

334. Line-surveying 225 

335. Traversing 226 

336. Use of compass 227 

337. Banging out lines 227 

338. Farm-surveying 223 

339. With the engineer's chain 230 

340. Plattinsr 230 



341. 

342. 
343. 

344. 
345. 



346 
847, 
348, 
349. 
351, 



The Gradiexter. 

Description 230 

To establish grades 232 

To measure distances 232 

On sloping ground 233 

General directions. 235 

The Stadia. 

Description and use 235 

Formulas 238 

Description of tables 238 

General directions 239 

352. Examples of surveys 240 



Chapter V. — Obstacles in Angular Surveying. 



354-358. Perpendiculars 242 

359, 360. Parallels 244 

Obstacles to Alixe^iext. 
A. To prolong a Line, 

361. General method 245 

362. By perpendiculars 245 

363. By an equilateral triangle 245 

364. By triangulation 245 

365. When the bine is inaccessible 246 

SG6. With onlv an angular instrument 246 



B. To interpolate Points oa a Line. 

367. General method 24*5 

363. By a random line i47 

369, 370. By latitudes and departur. - »4£ 

371. By similar triangles 249 

372. By triangulation 219 

Obstacles to MEASUREMrxr. 
A. W7itn Both Ends of the Line an acces- 
sible. 

373. Previous means 848 



ANALYTICAL TABLE OF CONTENTS. 



543 



ARTICLE PAGE 

374. By triangulation 250 

375. By angles to known points 250 

B. When One End of 'the Line is inaccessible. 

376. By perpendiculars 250 

377. By equal angles 250 

378. By triangulation 250 

379. When one point can not be seen 

from the other 251 

380. From a point to an inaccessible line 251 

C. When Both Ends of the Line are inac- 



3S1. General method 251 

382-388. Problems 252 



ARTICLE 5W.E 

To SUPPLT OMISSIONS. 

General statement 257 

When length and bearing of a side 
are wanting 258 

When length of one side and bear- 
ing of another are wanting, and 
the deficient sides are adjacent. . 258 

When they are not adjacent 259 

When the lengths of two adjacent 
sides are wanting 260 

When they are not adjacent 261 

When the bearing of two adjacent 
sides are wanting 262 

When they are not adjacent 262 



389 
390 



391 



392. 
393. 



394 
395 



396. 



Chapter VI. — Laying out, parting off, and dividing up Land 
Latino out Land. 

397. Its nature 263 

398. To lay out squares 263 



421. 

422. 



To part off a quadrilateral 278 

To part off any figure 278 



399. To lay out rectangles 264 

400. To lay out triangles 264 

401. To lay out circles 265 

402. Town-lots 265 

403. Land sold for taxes 266 

404. New countries 266 

Parting off Land. 

405. Its object 267 

A. By a Line parallel to a Side. 

408. To part off a rectangle 267 

407. To part off a parallelogram 267 

408, 409. To part off a trapezoid 268 

- B. By a Line perpendicular to a Side. 

410. To part off a triangle 269 

411. To part off a quadrilateral 270 

412. To part off any figure 270 

G. By a Line running in any Given Direc- 
tion. 

413. To part off a triangle 271 

414. When the bearings are given 271 

415. To part off a quadrilateral 271 

416. To part off any figure 272 

D. By a Line starting from a Given Point 

in a Side. 

417. To part off a triangle 272 

418. To part off a quadrilateral 273 

419. To part off any figure 273 

E. By a Line passing through a Given 

Point within the Field. 

420. To part off a triangle 275 



F. By the Shortest Possible Line. 

423. To part off a triangle 280 

G. Land of Variable Value. 

424. Methods 281 

425. Straightening crooked fences 282 



426. 



Dividing up Land. 
Arrangement 



284 



284 
285 



Division of Triangles. 

427. By lines parallel to a side 

428. By lines perpendicular to a side . 

429. By lines running in any given di 

rection 

430. By lines starting from an angle. , 

431. By lines starting from a point in a 

side 285 

432. 433. By lines passing through a 

given point within the triangle.. 287 
434-436. Graphical solutions 289 

437. By the shortest line 290 

Division of Rectangles. 

438. By lines parallel to a side 



291 



Division of Trapezoids. 
439, 440. By lines parallel to the bases. 291 

441. By lines starting from points in a 

side ' 293 

442. Other cases 293 

Division of Quadrilaterals. 

443. By lines, parallel to a side 294 

444. By lines perpendicular to a side. . 296 

445. By lines, in any given direction. . . 296 



544: 



ANALYTICAL TABLE OF CONTENTS. 



ARTICLE PAGE 

446. By lines starting from an angle. . 296 

447. By lines starting from points in a 

side 296 

44S, 449. Graphical solutions 297 

Division of Polygons. 
450. By lines running in any direction. 298 



A.RTICLE PAGE 

451. By lines starting from an angle. . 299 

452. By lines starting from a point in 

a side 299 

453. By lines passing through a point 

within the figure 299 

454. Other problems 299 



Chapter VII.— Public-Lands Survey. 



455. General system 

456. Difficulty 

457. Bunning township-line= 

458. Bunning section-lines . . 

459. Exceptional methods . . . 

460. Marking-lines 

461. Marking-corners 

462. Field-books 



463. 
464. 
465. 



467. 
468. 
469. 
470. 

471. 



The Solar Cojipass. 

Use of instrument 

Definitions 

Description of instrument. . 

Adjustments. 
Order 

First adjustment 

Second adjustment. - 

Third adjustment 

Fourth adjustment 

Field- Work. 
General statement 



301 
302 
304 
306 
308 
311 
311 
315 



319 
319 
321 

324 
324 
324 
325 
325 

325 



472. Declination 325 

473. Eefraction 326 

474. To determine latitude 327 

475. To determine the meridian 328 

476. Bunning lines 328 

477. Use of magnetic needle 328 

478. Solar attachment 328 

479. Adjustments 330 

480. Adjustment of polar axis 330 

481. Adjustment of hour-are 330 

482. Use 332 

483. Saegmuller's solar apparatus 332 



To locate a Parallel of Latitude. 



484. First method 333 

485. Otherwise 334 

486. Approximate method 334 

4S7. Example 334 

488. Spheroidal formula 335 

489. Length of parallels 335 

| 490. Convergence of meridians 336 



PART II. 

LEVELING. 

Introduction. 

491. Leveling in general. 337 I 493. Indirect leveling 338 

492. Direct leveling 337 i 494. Barometric leveling 338 



Chapter I. — Direct Leveling. 



495. Leveling instruments 339 

496. Methods of operation 339 

497. Curvature 340 

498. Eefraction 341 

Perpendicular Levels. 

499. Principle 341 

500. Plumb-line levels 341 

501. Beflectino: levels 342 



Water- Levels. 

502. Continuous water-levels 344 

503. Visual water-levels 344 

Spirit- Levels. 

504. The bubble-tube 345 

505. Sensibility 345 

506. Block-level 346 

507. Level with sights 347 



ANALYTICAL TABLE OF CONTENTS. 



545 



ARTICLE PAGE 

508. Hand-reflected level 347 

509. Gurley's hand-level 348 

510. The telescopic level 349 

511. TheY-level 349 

512. The telescope 350 

513. The cross-hairs 350 

514. The level 351 

515. The supports 351 

516. The parallel plates 352 

517. Description of cross-section 353 

Adjustments. 

518. General statement 354 

519. First adjustment 354 

520. Second adjustment 355 

521. Third adjustment 356 

522. Centering the object-glass and 

eye-piece 356 

523. The " peg-method" of adjustment 357 

524. Egault's level 358 

525. Troughton's level 358 

526. Gravatt's level 359 

527. Lenoir's level 359 

528. Tripods 359 

529. Rods 359 

530. Targets 360 

531. Vernier 361 

532. The New York rod 361 

533. The Boston rod 362 

534. The Philadelphia rod. .... 362 

535. Speaking-rods „ 362 

The Practice. 

536. Field routine 364 

537. Field-notes 366 

538. First form of field-book 366 



ARTICLE PAGE 

539. Second form of field-book 369 

540. Third form of field-book 371 

541. Best length of sights 372 

542. Equal distances of sight 372 

543. Datum-level 372 

544. Bench-marks 373 

545. Check-levels 373 

546. Limits of precision 374 

547. Trial-levels 374 

548. Leveling for sections 374 

549. Profiles 374 

550. Cross-levels 375 

Difficulties. 

551. Steep slopes 376 

552. When the rod is too low 377 

553. When the rod is too high 377 

554. When the rod is too near 378 

555. Water 378 

556. A swamp 378 

557. Underwood 379 

558. Board fence 379 

559. A wall 379 

560. A house 380 

561. The sun 380 

562. Wind 380 

563. Idiosyncrasies 380 

564. Eeciprocal leveling 381 

Leveling Location. 

565. Its nature 381 

566. Difficulties 382 

567. Staking out work 382 

568. To locate a level-line 383 

569. Applications 383 

570. To run a grade-line 384 



Chapter II. — Indirect Leveling. 



571. Vertical surveying 385 

572. Vertical angles 386 

573. Instruments 387 

574. Slopes 387 

575. Angular profiles. 388 

576. Burnier's level 388 

577. German universal instrument 389 

Simple Angular Leveling. 

A. For Short Distances. 

578. Principle 389 

579. Best-conditioned angle 389 

B. For G-reater Distances. 

580. Correction for curvature 3-90 

581. Correcting the result 390 



582. Correcting the angle 

583. Correcting for refraction . 



390 
391 



C. For Very Great Distances. 

584. Correction for curvature 391 

585. Correction for refraction 392 

586. Eeciprocal observations 392 

587. 588. Reduction to the summit of 

signals 394 

589. Leveling by the sea-horizon 395 

Compound Angular Leveling. 

590. General statement 397 

591. By angular co-ordinates in one plane 397 
592, 593. By angular co-ordinates in 

several planes 397 



546 



ANALYTICAL TABLE OF CONTEXTS. 



Chapter III. — Barometric Leveling. 



A ETICLE PAGE 

594. Principles 399 

595. Applications 899 

596. Correction for temperature of mer- 

cury 400 

597. Correction for temperature of air. . 400 

598. Other corrections 400 

599. Eules 400 

600. Formulas 401 

601. Correction for latitude and 

height 401 



602. English formula 402 

603. French formula 402 

604. Babinet's formula 403 

605. Tables 403 

606. Approximations 403 

607. Mountain barometer 404 

608. Aneroid barometer 404 

609. Hypsometer 405 

610. Accuracy of measurement 406 

611. Method of observing 406 



PAET III. 



612. Definition 



TOPOGRAPHY. 

Introduction. 
407 | 613. Systems. 






Chapter I. — First System. 



614. General ideas 403 

615. Plane of reference 409 

616. Vertical distance of sections 409 

617. Methods of determining contour- 

lines 409 

First Method. 

61S. General method 409 

619. On a narrow strip 410 

620. On a "broad surface 410 

621. Surverincr contour-lines 410 



Second Method. 
622. General method 



623. Irregular ground 411 

624. On a single hill. 411 

625. An extensive survey 412 

626. Interpolation 412 

627. Interpolating with the sector 412 

628. Eidges and thalwegs 413 

629. Forms of ground 414 

630. Sketching ground by contours 415 

631. Ambiguity 415 

632. Conventionalities 415 

633. Applications of contour-lines 416 

634. Sections by oblique planes 416 



410 



Chapter II. — By Lines of Greatest Slope. 



635. Their direction 417 

636. Sketching 4tf 



Details 417 



Chapter III. — By Shades from Vertical Light. 



638. Degree of shade 419 

639. Shades by tints 419 

640. Shades by contour-lines 419 

641. Shades by hatchings 420 

642. French method 420 



643. German method 420 

644. Diapason of tints 421 

Fourth System. 

645. By shades from oblique light 422 



ANALYTICAL TABLE OF CONTENTS. 



547 



Chapter IV. — Conventional Signs. 



ARTICLE PAGE 

646. Signs for natural surface 423 

647. Signs for vegetation 423 

648. Signs for water 426 

649. Colored topography 427 

650. Signs for miscellaneous objects... 428 

651. Scales 429 

The Plane-Table. 

652. General description 431 



653. The table. 



432 



ARTICLE PAGE 

654. The alidade 432 

655. Standard form of table 433 

656. Method of radiation 433 

657. Method of progression 435 

658. Method of intersection 436 

659. Method of resection 437 

660. To orient the table 438 

G61. To find one's place on the ground 439 

662. Inaccessible distances 440 

663. Contouring 440 



PART IV. 



TRIANG ULAR S UR V EYING. 
Chapter I. — Plane Surfaces. 



664. Method. .' 442 

665. Outline of operations 442 

666. Measuring a base 443 

667. Measuring with rods 443 

668. Measuring with a steel tape 445 

669. Corrections of base 446 

670. Keducing base to level of sea 446 

671. A broken base 447 

672. To interpolate a base 448 

673. Base of verification ... 449 

674. Choice of stations 450 



675. Signals 452 

676. Observation of angles 455 

677. Eeduction to center 456 

678. Correction of angles 458 

678 1 . Calculation and platting 458 

679. Interior filling up 459 

680. Radiating triangulation 459 

681. Farm triangulation 460 

682. Inaccessible areas 460 

683. Inversion of the fourth method... 461 

684. Defects of method of intersection. 401 



Chapter II.— Spherical Surveying. 



685. 



687. 
688. 



690. 



Nature 462 

Eeconnaissance 462 

The base 463 

The angles 463 

Computation of the triangles 463 

Spherical excess 464 



691. Adjustment of angles. 



465 



692. Legendre's theorem 466 

693. Accuracy of work 467 

694. Adjustment of a triangulation 467 

695. Co-ordinates of points 468 

696. 697. Problems 469 

698. References 470 



548 



ANALYTICAL TABLE OF CONTENTS. 



PAKT V. 



HYDR GRAPHICAL S UR VEYINO. 



ARTICLE 

699. Object 



PAGE 

. 471 



Chapter I. — The Sextant. 



PAGE 

700. Principle •. 472 

701. Description 473 

702. Box-sextant 474 

703. Beflecting circle 474 

704. Adjustments 474 

705. How to observe 475 

706. To set out perpendiculars 476 

707. Optical square 476 

708. 709. To measure a line, one end 

inaccessible 477 



710. To measure an inaccessible line. . 478 

711. Obstacles 479 

712. To measure heights 479 

713. To observe altitudes 480 

714. Sun's limbs 480 

715. Small altitudes and depressions.. 481 

716. Slopes 481 

717. Oblique angles 482 

717 1 . Advantages of sextant 483 



Chapter II. — Trilinear Surveying. 



718. Method 485 

Problem of the Three Points. 

719. Geometrical solution 485 



720. Instrumental solution 487 

721. Analytical solution 4S7 



Chapter III. — Surveying the Shore-Line. 

722. High-water line 489 ! 724. Measuring a base on water 490 

723. Low-water line 489 j 



Chapter TV. — Soundings. 



725. Object 491 

726. Bod and line 491 

727. Marking stations 491 

Determining Points on the Water. 

729. From the shore 492 

730. From boat with compass 492 

731. From boat with sextant 493 



732. Between stations 493 

733. In a river 493 

734. On a sea-coast 494 

735. Tide-gauges 494 

736. Establishment of a port 494 

737. Gauges in rivers 495 

738. Beacons and buovs 495 



T 39. Methods. 



Chapter V. — The Chart. 

496 I 740. Conventional signs. 



m 



PAET VI. 

UNDERGROUND OR MINING SURVEYING. 
Chapter I. — Surveying Old Lines. 



742. Surveying present workings . 

743. Difficulties 



49S 
499 



744. Stations 499 

745. Marking stations ? 499 



ANALYTICAL TABLE OF CONTENTS. 



549 



ABTICLE PAGE 

746. Transit-points 501 

747. Giving the sights 501 

748. Mining-transit 502 

749. Taking the sights 503 

750. Measuring the angles 503 

751. Plumbing the shaft 504 

752. Suspending the wires 505 



ARTICLE PAGE 

753. Setting the instrument in line 507 

754. Second method 507 

755. Third method 508 

756. Fourth method 508 

757. Keeping the notes 509 

758. Tabling the survey 513- 

759. Making the map 516 



Chapter II. — Locating New Lines. 



760. Second object... 518 

761. When mine is entered by an adit.. 518 

762. When mine is entered by a shaft. 518 

763. To dispense with the magnetic 

needle 519 



764. Eepeating the underground 

courses 520 

765. Third object 521 

766. Problems 521 



APPENDIX. 

APPENDIX A. 
SYNOPSIS OF PLANE TRIGONOMETRY. 



1. Definition .....: 523 

2. Angles and arcs 523 

3. Trigonometrical lines 524 

4. The lines as ratios 525 

5. Their variations in length 525 

6. Their changes of sign 526 



7. Their mutual relations 527 

8. Two arcs 528 

9. Double and half arcs 528 

10. The tables 529 

11. Eight-angled triangles 529 

12. Oblique-angled triangles 530 



APPENDIX B. 



Theory of transversals 

Harmonic division 534 



TRANSVERSALS, ETC. 

532 I The complete quadrilateral. 



536 



TABLES. 

Traverse-tables. 

Table of chords. 

Logarithms of numbers. 

Logarithmic sines, cosines, tangents, etc. 

Natural sines, cosines, tangents, etc. 

Stadia-table. 

Table of refraction in declination. 



TRAVERSE TABLES: 



OR, 



LATITUDES AND DEPARTURES OF COURSES 



CALCULATED TO 



THREE DECIMAL PLACES: 



EACH QUARTER DEGREE OF BEARING 



m 









LATITUDES - 


fcND DEPARTURES. 




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LATITUDES AND DEPARTURES. 




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LATITUDES 


AND 


DEPARTURES. 


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6i3 


i.i83 


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1-774 


3-226 


2.365 


4-o32 


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36£ 


o-8o4 


0-595 1 


608 


1. 190 


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r-784 


3-2i5 


2.379 


4-019 


53* 


36| 


o-8oi 


0.598 1 


6o3 


i-i97 


2.404 


1-795 


3 V 2o5 


2-3 9 3 


4- 006 


53i 


37° 


0-799 1 °*6o2 1 


5 97 


1.204 


2.396 


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3 • 1 95 


2-407 


3.993 


53° 


37i 


0-796 


o«6o5 1 


392 


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0.609 1 


587 


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3-i 7 3 


2-435 


3.967 


52* 


3?! 


0-791 


0«6l2 I 


58 1 


1-224 


2-372 


i-83 7 


3-i63 


2-449 


3.953 


524 


3§° 


0-788 


0-616 1 


5 7 6 


I«23l 


2-364 


1-847 


3-i52 


2-463 


3-940 


52° 


384 


o-785 


0.619 l 


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1-238 


2-356 


i-85 7 


3-i4i 


2-476 


3-927 


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38£ 


0.783 


0-623 1 


565 


1*245 


j 2-348 


1-868 


3-i3o 


2-490 


3.913 


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38| 


0*780 


0-626 1 


56o 


I -252 


1 2-34o 


1-878 


3- 120 


2-5o4 


3.899 


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39° 


0.777 


0-629 * 


554 


1.259 


2-33x 


j -888 


3-109 


2-5i7 


3-886 


51° 


3 9 i 


0-774 


o-633 1 


549 


1-265 


2-323 


1-898 


3.098 


2-53i 


3-872 


5oJ 


3 9 i 


0-772 


o-636 1 


543 


1.272 


2-3i5 


1-908 


3.086 


2-544 


3-858 


5o* 


391 


0-769 


0-639 1 


538 


1.279 


2- 307 


1-918 


3-o 7 5 


2-558 


3-844 


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40° 


0.766 


0-643 1 


532 


1.286 


2.298 


1.928 


3.064 


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o. 7 63 


• 646 1 


526 


1 .292 


i 2-290 


1.938 


: 3.053 


2084 


3-8i6 


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4H 


0-760 


- 649 1 


521 


1.299 


1 2.281 


1-948 


3-042 


2098 


3.802 


49i 


4q| 


o-758 


o-653 1 


5i5 


i-3o6 


2 273 


1.938 


3-o3o 


2.611 


3.-^ 


49\ 


41° 


o- 7 55 


o.656 1 


509 


I-3I2 


2 264 


1.96S 


. 3-019 


2.624 


3-774 


49 ~ 


4*i 


0*752 


0.659 l 


5o4 


1 .319 


2 256 


1.978 


3.007 


2.63 7 


3.759 


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4ii 


0-749 ! o.663 1 


498 


1 -325 


2-247 


1.988 


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2.65o | 


48* 


4i| 


- 746 . 666 [ 


492 


1-332 


2-238 


1.998 


2.984 


2.664 : 


3- 7 3o 


4»4 


42° 


0-743 


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486 


1-333 


2.229 


2.007 


2.973 


2.677 


3.716 


48° 


424 


o-74o 


0.672 1 


48o 


i,345 


2-221 


2.017 


2.961 


2.689 


3.701 


47|- 


42i 


0.737 


0.676 1 


475 


i-35i 


2-212 


2.027 


2-949 


2.702 


3-686 


47i 


4*1 


0-734 


0.679 l 


46 9 


1-358 


2>2o3 


2.036 


2-937 


2.715 


3-6^2 


474 


43° 


o-73i 


0-682 1 


463 


1-364 


2«I04 


2.046 


2-925 


2.728 


3-65 7 


47° 


43i 


o-7i8 


0-685 1 


45 7 


i»3 7 o 


2-185 


2.o56 


2-913 


2.741 


3-642 


46| 


43i 


0' 725 


0-688 1 


45i 


i-3 77 


2.I76 


2-o65 


2-901 


2-753 


3.627 


46± 


43| 


o- 722 


0-692 1 


445 


1-383 


2.167 


2.075 


2-889 


2.-66 


3.612 


461 


44° 


0.719 


0-695 1 


43 9 


1.389 


; 2 .i58 


2.084 


2-5" 


2-779 


3- 5 97 


46° 


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0-698 1 


433 


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2.149 


2.093 


2-865 


2.791 


3-582 


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0.701 1 


427 


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2-io3 


a-853 


2- 804 


3-566 


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0.710 


0.704 1 


420 


1.408 


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454 


45° 


0-707 


0.707 1 


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2«12I 


2-828 


2-828 


3-536! 


45 c 

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Lat. 


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2 


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S 



LATITUDES AND DEPARTURES. 


g 1 

Cfe* 

30° 


3 ! e 


•7 


8 


O 


<=>h 

60° 


Dep. 
2-5oo 


Lat. 


Dep. 


Lat. 
6-062 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


5.196 


3-000 


3-5oo 


6-928 


4- 000 


7-794 


4-5oo 


3oJ 


2-519 


5-i83 


3>023 


6-o47 


3- 


526 


6-91 1 


4-o3o 


7- 


775 


4-534 


59! 


3oi 


2-538 


5-170 


3-o45 


6-o3i 


3 


553 


6- 8 9 3 


4-060 


7- 


7 55 


4-568 


5 9 i 


3o| 


2-556 


5-i56 


3.o68 


6.016 


3 


579 


6-8 7 5 


4-090 


7 


7 35 


4-602 


591 


31° 


2-5 7 5 


5-i43 


3-090 


6- 000 


3- 


6o5 


6-85 7 


4- 120 


7 


7i5 


4-635 


59° 


3M 


2-594 


5-129 


3-n3 


5.984 


3 


63 1 


6-83 9 


4-i5o 


1 


694 


4-669 


58| 


3M 


3-6l2 


5- 116 


3-i35 


5.968 


3 


65 7 


6-821 


4 80 


7 


6 7 4 


4-702 


58i 


3i| 


2-63i 


5-102 


3-i5 7 


5- 9 52 


3 


683 


6-8o3 


4-210 


7 


653 


4-736 


584 


32° 


2-65o 


5-088 


3-i8o 


5- 9 36 


3 


709 


6.784 


4-239 


T 


632 


4-769 


58° 


32^ 


2-668 


5-074 


3- 202 


5-920 


3 


735 


6.766 


4-269 


7 


612 


4- 802 


57 1 


32^ 


2-686 


5.060 


3.224 


5.904 


3 


761 


6-747 


4-298 


7 


5 9 i 


4-836 


5 7 * 


3 2 | 


2-7o5 


5-o46 


3-246 


5-887 


3 


787 


6-728 


4-328 


7 


569 


4-869 


5 7 4 


33° 


2-723 


5-o32 


3.268 


5-87i 


3 


812 


6-709 


4-35 7 


7 


548 


4-902 


57° 


334 


2.741 


5-oi8 


3-290 


5-854 


3 


838 


6-690 


4-386 


7 


5^7 


4-935 


56} 


334 


2.760 


5-oo3 


3-3i2 


5-83 7 


3 


864 


6-671 


4-4i6 


7 


5o5 


4-967 


564 


334 


2.778 


4-989 


3-333 


5-820 


3 


889 


6-652 


4-445 


7 


483 


5- 000 


564 


34° 


2-796 


4-974 


3-355 


5-8o3 


3 


914 


6-632 


4-474 


7 


46i 


5-o33 


56° 


344 


2.814 


4-960 


3-3 77 


5-786 


3 


940 


6-6i3 


4-5o2 


-7 


439 


5-o65 


55| 


34* 


2-832 


4-945 


3- 3 9 8 


5.769 


3 


9 65 


6-593 


4-53i 


7 


4i 7 


5-098 


554 


34| 
35° 


2-85o 


4-930 


3-420 


5.752 


3 


990 


6-5 7 3 


4-56o 


7 


3 9 5 


5-i3o 


554 
55° 


2-868 


4- 9 i5 


3-44i 


5.734 


4 


oi5 


6-553 


4-58 9 


7 


3 7 2 


5. 162 


354 


2-886 


4.900 


3-463 


5.716 


4 


o4o 


6-533 


4-6i 7 


7 


35o 


5.194 


54} 


35* 


2 • 904 


4-885 


3-484 


5-699 


4 


o65 


6-5i3 


4-646 


7 


327 


5.226 


544 


35| 


2-921 


4-869 


3-5o5 


5.68i 


4 


090 


6.493 


4-6 7 4 


7 


3o4 


5-258 


544 


36° 


2- 9 3 9 


4-854 


3-527 


5-663 


4 


1 1 5 


6.472 


4-702 


7 


281 


5-290 


54° 


364 


2-957 


4-83 9 


3-548 


5-645 


4 


i3 9 


6-452 


4-73o 


7 


258 


5-322 


53} 


36£ 


2-974 


4-823 


3.569 


5-627 


4 


1 64 


6.43i 


4-7^9 


7 


235 


5-353 


534 


36| 


2-992 


4- 808 


3.590 


5-609 


4 


188 


6-4io 


4-787 


7 


211 


5-385 


534 


31° 


3-009 


4-79 2 


3. 611 


5-590 


4 


2l3 


6.389 


4-8i5 


7 


188 


5.4i6 


53 G 


3?i 


3-026 


4-776 


3-632 


5-5 7 2 


4 


207 


6.368 


4-842 


7 


1 64 


5-448 


52} 


3?4 


3-044 


4-760 


3-653 


5.554 


4 


261 


6.347 


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7 


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5.479 


524 


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4-744 


3-6 7 3 


5-535 


4 


286 


6.3 2 6 


4-898 


7 


116 


5-5io 


52.4 


3§° 


3-078 


4.728 


3- 694 


5-5i6 


4 


3io 


6-3o4 


4-925 


7 


092 


5.54i 


52° 


384 


3-095 


4-712 


3.7i5 


5-497 


4 


334 


6.283 


4-953 


7 


068 


5.572 


5i} 


384 


3-n3 


4-696 


3-735 


5-478 


4 


358 


6.261 


4-980 


7 


o43 


5-6o3 


5i4 


38| 


3-i3o 


4-679 


3-756 


5.459 


4 


38 1 


6.239 


5 -007 


7 


019 


5-633 


5i4 


39° 


3-i47 


4-663 


3.776 


5-44o 


4 


4o5 


6-217 


5-o35 


6 


994 


5-664 


51° 


M 


3- 164 


4-646 


3.796 


5-42i 


4 


429 


6-i 9 5 


5.062 


6 


970 


5-6 9 4 


5o} 


3 9 i 


3-i8o 


4-63o 


3-8i6 


5-4oi 


4 


453 


6-i 7 3 


5-089 


6 


945 


5-725 


5o4 


39l 
40° 


3-197 


4-6i3 


3-83 7 


5-382 


4 


476 


6-i5i 


5-n6 


6 


920 


5-755 


5o4 
50° 


3.2i4 


4- 5 9 6 


3-85 7 


5-362 


4 


5oo 


6.128 


5.142 


6 


8 9 4 


5. 7 85 


4o4 


3. 2 3i 


4-579 


3.877 


5-343 


4 


523 


6« 106 


5.169 


6 


869 


5-8i5 


49* 


4<>i 


3-247 


4-562 


3.897 


5-3 2 3 


4 


546 


6-o83 


5> 196 


6 


844 


5-845 


49i 


4o| 


3.264 


4-545 


3.917 


5-3o3 


4 


56 9 


6.061 


5.222 


6 


818 


5.875 


49^ 


41° 


3.280 


4-528 


3- 9 36 


5-283 


4 


592 


6-o38 


5-248 


6 


•792 


5.905 


49° 


4«* 


3.297 


4-5ti 


3- 9 56 


5-263 


4 


6i5 


6- 01 5 


5.275 


6 


767 


5.934 


48} 


4ii 


3.3i3 


4.494 


3.976 


5.243 


4 


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5.992 


5-3oi 


6 


74l 


5.964 


484 


4r| 


3.329 


4.476 


3.995 


5-222 


4 


661 


5.968 


5-327 


6 


-7i5 


5.993 


484 


42° 


3-346 


4- 45 9 


4-oi5 


5«202 


4 


-684 


5.945 


5-353 


6 


688 


6«022 


4§° 


4H 


3-362 


4-44i 


4-o34 


5.182 


4 


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5.922 


5.379 


6 


662 


6-o5i 


47} 


424 


3.378 


4-424 


4-o54 


5- l6l 


4 


729 


5.898 


5-4o5 


6 


635 


6-080 


474 


4H 


3.394 


4- 406 


4-073 


5-l40 


4 


752 


5.875 


5-43o 


6 


609 


6> 109 


474 


43° 


3-4io 


4-388 


4-092 


5.1I9 


4 


■774 


5-85i 


5-456 


6 


■582 


6-i38 


47° 


434 


j 3.426 


4- 3 7 o 


4- hi 


5-O99 


4 


•796 


5.827 


5-48i 


6 


■555 


6-167 


46} 


4H 


i 3.442 


4-352 


4-i3o 


5.078 


4 


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5-8o3 


5«5o7 


6 


■ 528 


6-195 


464 


43* 


3-458 


4-334 


4-149 


5.057 


4 


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5-779 


5-532 


6 


5oi 


6-224 


464 


44° 


3.473 


4-3i6 


4-168 


5-035 


4 


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5.755 


5.557 


6 


474 


6-252 


46° 


444 


3-489 


4-298 


4-187 


5-oi4 


4 


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5-73o 


5-582 


6 


447 


6-280 


45} 


444 


; 3-5o5 


4-280 


4- 206 


4- 99 3 


4 


• 906 


5 • 706 


5.607 


6 


419 


6-3o8 


454 


44* 


3«52o 


4-261 


4-224 


4-97i 


4 


.928 


5-681 


5-632 


6 


392 


6-336 


454 


45° 

CD 

s* 

era 


3-536 


4-243 


4-243 


4-95o 


4- 95o 


5-65 7 


5-65 7 


6-364 


6-364 


45° 

tab 
oS 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


5 


6 


■7 


8 


O 



TABLE OF CHORDS: [R 


ADIU8 ~ 


= 1.0000J. 




1L 

o' 


0° 


1° 


2° 


3° 


40 


5° 

.0872 


6° 


7° 


8° 


9° 


10° 


M. 

O' 


.0000 


.0175 


.0349 


• o524 


.0698 


.1047 


• 1221 


• i395 


.1569 


•1743 


I 


• ooo3 


.0177 


• o352 


•0526J.0701 


• o8 7 5 


• io5o 


.1224 


• i3 9 8 


• 1D72 


•1746 


I 


2 


.0006 


.0180 


.o355 


.0529 .0704 


.0878 


• io53 


.1227 


• i4oi 


• i5 7 5 


•1749 


2 


3 


.0009 .oi83 


.o358 


.o53 2 


.0707 


.0881 


• io55 


• I23o 


• i4o4 


• i5 7 8 


.1752 


3 


4 


.00121.0186 


. .o36t 


• o535 


.0710 


.0884 


• io58 


.1233 


•i4o 7 


• i58i 


.i 7 55 


4 


5 


• ooi5 .0189 


!.o364 


• o538 


.0713 


.0887 


• 1061 


.1235 


• i4io 


.1584 


• i 7 58 


b 


6 


•00171.0192 


|.o366 


• o54i 


.0715 


.0890 


.1064 


.1238 


• i4i3 


.i58 7 


.1761 


6 


7 


•0020 -0195 


S .0369 


• o544 


.0718 


.0893 


• 1067 


.1241 


• i4i5 


• i58 9 


.1763 


7 


8 


•0023; '0198 


1.0372 


•o547 


• 0721 


.0896 


.1070 


•1244 


• i4i8 


.1592 


.1766 


8 


9 


•OO26: "020I 


|.o3 7 5 


• o55o 


.0724 


.0899 


.ic 7 3 


.1247 


.1421 


• i5 9 5 


.1769 


9 


IO 

- 


•OO29J '0204 


j.o3 7 8 


!.o553 


.0727 


• 0901 


.1076 


.125o 


.1424 


.1598 


.1772 


10 
11 


• oo32 


.0207 


|.o38i 


.o556 


• 0730 


.0904 


.1079 


.1253 


.1427 


.'601 


•1775 


12 


.oo35 


.0209 


• o384 


.o558 


.0733 


.0907 


.1082 


.1256 


• i43o 


.1604 


.1778 


12 


i3 


.oo38 


■ 0212 


• o387 


• c56i 


• 0736 


.0910 


.1084 


• 1259 


.1433 


. 1607 


.1781 


id 


i4 


• oo4i 


• 02l5 


; .0390 


• o564 


.0739 


.0913 


.1087 


• 1262 


• i436 


• 1610 


•1784 


i4 


i5 


•oo44i -0218 


, -0393 


• 0567 


.0742 


.0916 


.1090 


.1265 


• i43 9 


• i6i3 


•1787 


ib 


16 


.0047 • 022 1 


! • 0396 


• 0570 


•0745 


.0919 


.1093 


• 1267 


.1442 


.1616 


.1789 


16 


17 


.0049 -0224 


.0398 


• o573 


.0747 


.0922 


.1096 


.1270 


•i444 


.1618 


.1792 


17 


18 


•oo52 -0227 


1 • 040 1 


■ 0576 


.07^0 


.0925 


.1099 


.1273 


•i447 


• 162 1 


•I7<* 


18 


'9 


•oo55[ '023o 


• 0404 


•0579 


• o 7 53 


.0928 


• 1102 


• 1276 


• i45o 


.1624 


.1798 


J 9 


20 
21 


• oo58 


.0233 


• 0407 


• o582 


• 0756 
.0759 


.0931 


• no5 


.1279 


.i453 


. 1627 


.1801 


20 

21 


• 0061 


.0236 


• o4io 


• o585 


.o 9 33 


.1108 


.1282 


• i456 


• i63o 


.1804 


22 


• 0064 


.0239 


.o4i3 


• o^38 


• 0762 


.0936 


• in 1 


• 1285 


•1459 


.i633 


.1807 


22 


23 


.0067 


.0241 


• o4i6 


.0590 


• 0765 


.0939 


, .1114 


.1288 


.1462 


• i636 


.1810 


23 


24 


.0070 


.0244 


.0419 


.0593 


.0768 


.0942 


1 .1116 


• 1291 


• i465 


.i63 9 


• i8i3 


24 


25 


.0073'' 0247 


.0422 


.o5 9 6 


.0771 


.0945 


.1119 


.1294 


.1468 


.1642 


.1816 


2b 


26 


•OO76 '025o 


.0425 


.0599 


•0774 


.0948 


• 1122 


• 1296 


•i47i 


• i645 


.1818 


2* 


27 


•0079 «0253 


.0428 


.0602 


.0776 


.o 9 5 1 


• II2D 


•' 2 99 


•i473 


•1647 


.1821 


27 


28 


•0081 j '0256 


• o43o 


• o6o5 


•0779 


.0954 


.1128 


- i3o2 


.1476 


• i65o 


.1824 


28 


29 


•0084! -0259 


• o433 


.0608 


.0782 


• 0937 


• ii3i 


i3o5 


.1479 


.1653 


.1827 


29 


3o 
3i 


•0087 -0262 


• o436 


.061 1 


• o 7 85 


• 0960 


• u34 


• i3o8 


.1482 


• i656 


• i83o 


3o 
3i 


.0090 '0265 


.0439 


.0614 


.0788 


.0962 


• n3 7 


.i3ii 


• i485 


• 1659 


• i833 


32 


•0093 .0268 


.0442 


.0617 


.0791 


.0965 


.1140 


• i3i4 


.1488 


.1662 


• i836 


32 


33 


.0096: .0271 


• o445 


.0619 


•0794 


.0968 


• ii43 


• i3i 7 


•1491 


• i665 


.1839 


33 


34 


.0099 .0273 


• o448 


.0622 


.0797 


.0971 


• n45 


. r32o 


.i494 


.1668 


.1842 


34 


35 


•0I02 1 .0276 


• o45i 


.0625 


.0800 


.0974 


.1148 


• i323 


.1497 


.1671 


.i845 


3!) 


36 


•oio5| .0279 


•o454 


.0628 


• o8o3 


•0977 i 


.ii5i 


• i3a5 


. 1 5oo 


•1674 


.1847 


36 


37 


.0108! -0282 


•0457 


• o63i 


.0806 


.0980 


• 1.54 


.i3a8 


. i5o2 


.1676 


• i85o 


3 7 


38 


•OIII .0285 


• 0460 


• o634 


.0808 


.0983 


• I1D 7 


• i33i 


• i5o5 


1679 


.i853 


3s 


3 9 


• on3 .0288 


.0462 


• 0637 


.0811 


.0986 


.1160 


.i334 


• i5o8 


1682 


.i856 


3 9 


4o 
4i 


•0116 -0291 


• o465 


• o64o 


.0814 


. 0989 


• n63 


• i33 7 


• i5ii 


• i685 


.1859 


4o 
4i 


• 0119, .0294 


.0468 


.o643 


.0817 


.0992 


.1166 


.i34o 


• i5i4 


.1688 


.1862 


43 


•0122J .0297 


.0471 


.0646 


.0820 


.0994 : 


.1169 


.i343 


.i5i? 


• 1 691 


.1866 


42 


43 


.0125 «o3oo 


•o474 


.0649 


.0823 


.0997 


.1172 


.i346 


. l520 


.1694 


.1868 


43 


44 


.0128 -o3o3 


•0477 


• c55i 


.0826 


• 1 000 : 


.ii 7 5 


• i349 


.i523 


•1697 


.1871 1 


44 


45 


•oi3i «o3o5 


.0480 


• o654 


.0829 


• ioo3 ; 


.1177 


.i352 


.1526 


.1700 


•1873 ; 


45 


46 


.0134! «o3o8 


• o483 


.0657 


• o832 


• 1006 


.1180 


.i355 


• 1529 


. 1703 


.1876 


46 


47 


.0137 «o3ii 


.0486 


.0660 


• o835 


• 1009 


• n83 


■ i35 7 


• i53i 


. 1705 


.1879 


47 


48 


•oi4o -o3i4 


.0489 


.o663 


• o838 


• 1012 


.1186 


• i36o 


• i534 


.1708 


.1882 


48 


49 


•or43l'03i7 


.0492 


.0666 


■ o84o 


• ioi5 


.1189 


. i363 


• i53 7 


.1711 


• i885 


49 


5o 
5i 


• oi45 


• o320 


•0494 


.0669 


• o843 


.1018 


• 1 192 


.i366 


• i54o 


•i7i4 


.1888 


5o 
5i 


• oi48 


• o323 


•0497 


.0672 


.0846 


•1021 ! 


.1195 


• i36o 


. i543 


•1717 


•1891 : 


52 


• oi5i 


• o326 


• o5oo 


•0675] .0849 


• 1023 


.1198 


•l3 7 2 


• i546 


• 1720 


• 1894 


52 


53 


• 01 54 'o32-9 


•o5o3, 


.06781.0852 


• 1026 


• 1201 


• i3 7 5 


.1549 


• 1723 


• 1897 


53 


54 1 


•oi57J «o332 


• o5o6 


.0681 -o855 


• 1029 


• I204 


.[3-8 


. i552 


.1726 


. 1900 


54 


55 


.0160 «o335 


• 0509 


• o683 Lo858 


. io32 


. I206 


• i38i 


. i555 


•1729 


.1902 : 


56 


•oi63 «o337 


• o5i2 


.0686 .0861 


. io35 


. I209 


• i384 


• i55S 


• i 7 32 


. 1905 


56 


5 7 


•0166 'o34o 


• o5i5 


.0689 .0864 


• io38 


• 1212 


• i386 


• i56o 


.. 7 34 


.1908 


5- 


58 


• 0169! «o343 


• o5i8 


• 0692 


• 0867 


• io4i ! 


• I2l5 


• i3S 9 


• i563 


•i73 7 


.l 9 M 


58 


5 9 


•0172! o346 


• o52i 


.0695 


.0869 


. io44 
•io47 1 


• I2l8 


• 1392 


• i566 


- 


•1914 


^9 


60 


•0175-0349 


• o524 


.0698 


.0872 


• 1221 


• i3 9 5 


• i56 9 


• 1743 


• 1917 


00 



■'" ■'--■"■""' ! — '■ — ' ■ — ' ' 

TABLE OF CHORDS: [Radius = 1.0000]. 


M. 

o' 


11° 


12° 


13° 


14° 

.2437 


15° 


16° 

.2783 


17° 


18° 
.3129 


19° 


20° 


21° 


M. 
O 


.1917 


.2091 


.2264 


.261 1 


.2956 


.33oi 


- 3473 


- 3645 


I 


.1020 


.2093 


.2267 


• 2440 


• 26i3 


.2786 


.2959 


.3i32 


• 33o4 


- 34-6 


• 3648 


I 


2 


• 1923 


.2096 


.2270 


• 2443 


.2616 


.2789 


• 2962 


.3i34 


.33o<7 


• U-9 


• 3o5o 


2 


3 


.1926 


.2099 


.2273 


.2446 


.2619 


.2792 


.2965 


.3i3 7 


• 33io 


.3482 


. -(653 


3 


4 


.1928 


.2102 


.2276 


•2449 


.2622 


.2795 


.2968 


• 3i4o 


.33i2 


• 34H4 


. 3^56 


4 


5 


• 1 93 1 


• 2IOD 


.2279 


.2452 


.2625 


.2798 


.2971 


.3i43 


• 33i5 


• 34»7 


.3659 


5 


6 


.1934 


• 2108 


.2281 


• 2455 


.2628 


.2801 


.2973 


• 3i46 


• 33i8 


• 3490 


.3662 


b 


7 


.1937 


• 2III 


.2284 


• 2458 


• 263 1 


.2804 


.2976 


.3149 


.3321 


.3493 


.3665 


7 


8 


.1940 


• 2Il4 


.2287 


• 2460 


• 2634 


.2807 


•2979 


.3i5a 


.3324 


.3496 


• 3668 


8 


9 


.1943 


•21 17 


• 2290 


• 2463 


• 2636 


.2809 


• 2982 


• 3i55 


.3327 


• 3499 


.3670 


9 


IO 

ii 


• i 9 46 


•2II9 


.2293 


.2466 


.2639 


.2812 


• 2 9 85 


• 3i5 7 


..333o 


• 35o2 


• 36 7 3 


10 
11 


•1949 


•2122 


.2296 


.2469 


.2642 


.2815 


.2988 


• 3 160 


• 3333 


• 35o4 


.3676 


12 


.1952 


• 2125 


• 2299 


.2472 


.2645 


.2818 


2991 


• 3i63 


.3335 


.35o 7 


.3679 


12 


i3 


.1955 


• 2128 


• 2302 


•24 7 5 


.2648 


.2821 


.2994 


• 3i66 


• 3338 


.35io 


• 3682 


i-3 


i4 


.1957 


• 2l3l 


• 23o5 


.2478 


• 265 1 


.2824 


.2996 


.3169 


.334i 


• 35i3 


• 3685 


i4 


i5 


. i960 


.2134 


• 2307 


.2481 


• 2654 


.2827 


.2999 


• 3r72 


• 3344 


• 35i6 


• 3688 


i5 


16 


.1963 


•2l3 7 


•23lO 


• 2484 


• 265 7 


. 2 83o 


• 3oo2 


• 3i 7 5 


• 3347 


• 35i 9 


• 3690 


16 


17 


.1966 


• 2l4o 


• 23i3 


.2486 


.2660 


.2832 


• 3oo5 


.3178 


• 335o 


• 3522 


.3693 


17 


18 


.1969 


.2143 


. 2 3r6 


• 2489 


.2662 


.2835 


• 3oo8 


• 3r8o 


• 3353 


• 3525 


• 36 9 6 


18 


19 


.1972 


• 2146 


• 2319 


• 2.492 


• 2665 


• 2838 


• 3on 


3i83 


• 3355 


.3527 


• 3699 


»9 


20 
21 


.1975 


.2148 


•2322 


• 2495 


.2668 


.2841 


• 3oi4 


• 3i86 


• 3358 


• 353o 


• 3702 


20 
21 


.1978 


•2l5l 


•2325 


.2498 


• 2671 


.2844 


• 3017 


• 3i8 9 


• 336i 


.3533 


• 3 7 o5 


22 


.1981 


• 2i54 


• 2328 


• 25oi 


• 2,674 


.2847 


.3019 


.3192 


.3364 


.3536 


.3708 


22 


23 


• i 9 83 


•21 57 


• 233i 


• 2 5o4 


• 2677 


• 2 85o 


• 3o22 


• 3i 9 5 


• 336 7 


.3539 


.3710 


23 


24 


.1986 


• 2160 


• 2333 


• 2507 


.2680 


. 2853 


• 3o25 


• 3i 9 8 


• 3370 


.3542 


.37i3 


24 


25 


.1989 


• 2i63 


• 2336 


•25lO 


• 2683 


.2855 


.3028 


•3200 


.3373 


.3545 


• 3 7 i6 


25 


26 


.1992 


• 2166 


• 2339 


•25l2 


• 2685 


.2858 


• 3o3i 


• 32o3 


.3376 


•3547 


• 3719 


26 


27 


.1995 


• 2169 


.2342 


• 25i5 


.2688 


.2801 


• 3o34 


• 32o6 


.3378 


• 355o 


.3722 


27 


28 


.1998 


• 2172 


.2345 


• 25i8 


• 2691 


.2864 


• 3o37 


.3209 


• 338t 


• 3553 


• 3 7 25 


28 


29 


• 2001 


.2174 


.2348 


•2521 


•2694 


.2867 


• 3o4o 


.3212 


• 3384 


.3556 


.3728 


29 


3c 
3i 


•20S4 


.2177 


• a35i 


• 2524 


• 2697 


• 2870 


• 3o42 


• 32i5 


• 338 7 


.3559 


.3730 


3o 
3i 


.2007 


.2180 


.2.354 


.2527 


• 2700 


.2873 


• 3o45 


.3218 


.3390 


• 3562 


• 3 7 33 


32 


.2010 


.2183 


• 235 7 


• 253o 


-2703 


.2876 


• 3o48 


.3221 


• 33 9 3 


.3565 


• 3 7 36 


32 


33 


.2012 


.2186 


• 2 35 9 


.2533 


.2706 


.2878 


.3o5i 


• 3223 


.3396 


• 356 7 


• 3 7 3 9 


33 


34 


• 20l5 


.2189 


• 2362 


.2536 


• 2709 


.2881 


• 3o54 


.3226 


.3398 


.3570 


•3 7 42 


34 


35 


.2018 


.2192 


.2365 


• 2538 


.271 1 


.2884 


• 3o5 7 


.3229 


.3401 


• 35 7 3 


•3 7 45 


35 


36 


.2021 


• 2195 


• 2368 


• 2541 


.2714 


.2887 


• 3o6o 


.3232 


.34o4 


.3576 


• 3 7 48 


36 


3 7 


.2024 


.2198 


• 2371 


• 2544 


.2717 


.2890 


• 3o63 


.3235 


• 3407 


.3579 


• 3 7 5o 


37 


38 


• 2027 


.2200 


•23 7 4 


•2547 


• 2720 


.2893 


• 3o65 


.3238 


.3410 


• 3582 


• 3 7 53 


38 


39 


• 2o3o 


• 22o3 


• 23 77 


• 255o 


.2723 


.2896 


• 3o68 


.3241 


• 34i3 


.3585 


• 3 7 56 


3 9 


4c 
4i 


• 2o33 


• 2200 


• 238o 


• 2553 

• 2556 


• 2726 


.2899 


• 3071 


• 3244 


.3416 


• 358 7 


• 3 7 5 9 
.3762 


4o 
4i 


• 2o36 


• 2209 


• 2383 


.2729 


.2902 


.3074 


.3246 


.3419 


.3590 


42 


.2o38 


•2212 


• 2385 


.2559 


.2732 


.2904 


.3o 77 


.3249 


.3421 


• 35 9 3 


• 3 7 65 


42 


43 


. 204 1 


• 22 1 5 


• 2388 


.256i 


•2734 


.2907 


• 3o8o 


• 3252 


.3424 


.3596 


• 3 7 68 


43 


44 


.2044 


.2218 


.2391 


.2564 


.2-37 


.2910 


.3o83 


• 3255 


.3427 


.3599 


.3770 


44 


45 


.2047 


.2221 


.2394 


• 2567 


.2740 


.2913 


• 3o86 


.3258 


• 343o 


.3602 


.3 77 3 


45 


46 


• 2o5o 


.2224 


.2397 


.2570 


.2743 


• 2916 


• 3o88 


• 3 2 6i 


.3433 


• 36o5 


.3776 


46 


4" 


• 2o53 


.2226 


.2400 


.2573 


• 27/16 


.2919 


.3091 


.3264 


• M66 


• 36o8 


.3779 


47 


48 


• 2o56 


• 2229 


• 2.403 


• 2576 


•2749 


.2922 


.3094 


.3267 


.3439 


• 36io 


.3782 


48 


49 


.2059 


• 2232 


.2406 


.2579 


• 2752 


.2925 


.3097 


• 3269 


.344i 


• 36i3 


• 3 7 85 


49 


5o 
5i 


• 2062 

• 2o65 


• 2235 


• 2409 

• 241 1 


• 2582 


• 2 7 55 
.2 7 58 


.2927 


• 3ioo 


• 3272 


•3444 


• 36i6 


• 3 7 88 


5o 
5i 


• 2238 


• 2585 


.2930 


• 3io3 


• 32 7 5 


•3447 


.3619 


.3790 


52 


.2067 


.2241 


•2414 


• 258 7 


2760 


. 2 9 33 


• 3io6 


.3278 


• 345o 


.3622 


.3793 


52 


53 


.2070 


• 2244 


.2417 


.2590 


• 2 7 63 


.2936 


• 3109 


.8281 


.3453 


.3625 


.3796 


53 


54 


• 2073 


.2247 


• 24-20 


• 2593 


• 2766 


.2 9 3 9 


• 3m 


• 3284 


• 3456 


.3628 


•3799 


54 


55 


• 2( -6 


• 225o 


• 2423 


• 2.596 


.2769 


.2942 


• 3u4 


.3287 


• 345 9 


• 363o 


.3802 


55 


56 


• 2079 


•2253 


.2426 


.25 99 


• 2772 


.2945 


• 3ii7 


.3289 


.3462 


.3633 


• 38o5 


56 


57 


.2082 


• 2255 


• 2429 


• 2602 


• 2 77 5 


. 2948 


•3l20 


.3292 


• 3464 


.3636 


• 38o8 


5 7 


58 


• 2o85 


• 2258 


.2432 


• 26o5 


.2778 


.2950 


• 3i23 


• 3295 


.3467 


• 363 9 


• 38io 


58 


^9 


.2088 


• 2261 


.2434 


.2608 


.2781 


.2953 


.3126 


.3298 


.3470 


.3642 


• 38i3 


5 9 


00 


|.20 9 I 


.2264 


.2437 


• 261 1 


• 2 7 83 


.2956 


.3129 


• 33oi 


•347* 


I 3645 


• 38i6 


60 



TABLE OF CHORDS: [Radius = 1.0000]. 


M. 

©' 


22° 

• 38j6 


23° 

.3987 


24° 
• 4i58 


25° 

.4329 


26° 

.4499 


27° 
.4669 


28° 
• 4838 


29° 


30° 


31° 


32° 


M. 

O' 


.5oo8 


.5c 7 6 


.5345 


• 55i3 


I 


.38i 9 


.3990 


.4161 


• 4332 


• 45o2 


.4672 


.4841 


• 5oio 


• 5r79 


• 5348 


• 55i6 


I 


2 


.3822 


.3993 


.4164 


• 4334 


• 45o5 


• 46 7 5 


•4844 


• 5oi3 


.5182 


• 535o 


• 55i8 


2 


3 


.38?5 


.3996 


4167 


.4337 


• 45o8 


.4677 


•4847 


• 5oi6 


• 5i85 


• 5353 


• 5521 


3 


4 


.3828 


.3999 


.4170 


.434o 


• 45io 


.4680 


• 485o 


.5019 


• 5i88 


• 5356 


• 5524 


4 


5 


• 383o 


.4002 


.4172 


.4343 


• 45i3 


.4683 


• 4853 


• 5o22 


.5190 : 


• 535 9 


• 5527 


5 


6 


• 3833 


• 4oo4 


•4i75 


• 4346 


• 45i6 


.4686 


• 4855 


.5024 


.5193 


.5362 


■ 553o 


6 


7 


• 3836 


.4007 


•4178 


.4349 


•45i 9 


.4689 


• 4858 


.5027 


• 5196 


• 5364 


• 5532 


7 


8 


• 383 9 


• 4oio 


.4181 


• 4352 


.4522 


.4692 


.4861 


• 5o3o 


.5199 ' 


• 536 7 


• 5535 


8 


9 


• 3842 


• 4oi3 


.4184 


• 4354 


• 4525 


•46 9 4 


• 4864 


• 5o33 


• 5202 


• 53 7 o 


• 5538 


9 


IO 

ii 


• 3845 
.3848 


• 4oi6 


•4i8 7 


•435 7 
• 436o 


.4527 
• 453o 


•4697 


.4867 


• 5o36 

• 5o3 9 


•52o4 1 


• 53 7 3 


• 554i 

1 



1 1 


.4019 


.4190 


.4700 


.4869 


•5207 1 


• 53 7 6 


• 5543 


12 


.385o 


.4022 


.4192 


• 4363 


• 4533 


• 4 7 o3 


.4872 


• 5o4i 


• 52IO 


.5378 


.5546 


12 


i3 


.3853 


.4024 


•4i95 


• 4366 


• 4536 


.4706 


• 48 7 5 


• 5o44 


• 52i3 


• 538i 


• 5549 


i3 


i4 


.3856 


.4027 


.4198 


.4369 


.4539 


.4708 


.4878 


•5o47 


.5216 


.5384 


• 5552 


i4 


i5 


.385 9 


• 4o3o 


•4201 


•43 7 i 


.4542 


•47" 


.4881 


• 5o5o 


• 5219 


• 538 7 


• 5555 


i5 


16 


.3862 


• 4o33 


.4204 


•43 7 4 


•4544 


•47i4 


• 4884 


• 5o53 


•5221 j 


.5390 


•555 7 


16 


17 


.3865 


• 4o36 


• 4207 


.4377 


•4547 


.4717 


• 4886 


• 5o55 


•5224 ' 


.5392 


• 556o 


17 


18 


• 3868 


.4039 


.4209 


• 438o 


• 455o 


.4720 


.4889 


• 5o58 


• 5227 


• 53 9 5 


•5563 


18 


19 


.3870 


.4042 


.4212 


• 4383 


• 4553 


.4723 


.4892 


• 5o6i 


• 523o 


.5398 


•5566 


'9 


20 
21 


.3873 
• 38 7 6 


•4o44 
•4o4- 


• 42i5 
.4218 


• 4386 

• 4338 


• 4556 
•455 9 


• 4725 


• 48 9 5 


• 5o64 


• 5233 


• 54oi 

• 54o4 


• 556 9 
.55 7 i 


20 
21 


.4728 


.4898 


• 5o67 


• 5 2 35 


22 


.3879 


• 4o5o 


•4221 


.4391 


• 456i 


•473 1 


.4901 


• 5070 


•5 2 38 ; 


• 54o6 


•55 7 4 


22 


23 


.3882 


• 4o53 


•4224 


•43 9 4 


• 4564 


•4734 


.4903 


• 5072 


.5241 ; 


.5409 


•55 7 7 


23 


24 


.3885 


• 4o56 


•4226 


.4397 


•456 7 


.4737 


• 4906 


• 5o 7 5 


.5244 


.5412 


• 558o 


24 


25 


.3888 


.4059 


.4229 


• 44oo 


•45 7 o 


• 474o 


•4909 


• 5o 7 8 


•5247 


• 54i5 


• 5583 


25 


26 


• 38 9 o 


.4061 


•4232 


• 44o3 


•45 7 3 


• 4742 


.4912 


• 5o8i 


•5249 ' 


• 54i8 


• 5585 


26 


27 


• 38 9 3 


.4064 


•4235 


• 44o5 


•45 7 6 


•4745 


•49i5 


• 5o84 


• 5252 


.5420 


• 5588 


1 27 


28 


• 38 9 6 


.4067 


• 4238 


• 44o8 


•45 7 8 


•4748 


•4917 


• 5o86 


.5255 


• 5423 


• 5591 


28 


29 


.3899 


.4070 


• 4241 


•44n 


• 458i 


•475 1 


• 4920 


.5089 


.5258 


.5426 


•5594 


29 


3o 
3 1 


.3902 
.3905 


.4073 


•4244 


•44U 
.4417 


• 4584 
•458 7 


•4754 
•4757 


.4923 
.4926 


• 5092 

• 5o 9 5 


.5261 i 



.5263 


• 5429 

• 5432 


•55 97 
.5599 


3o 

3i 


.4076 


.4246 


32 


• 3 9 o8 


.4079 


.4249 


.4420 


.4590 


•4759 


.4929 


•5o 9 8 .5266 


.5434 


• 56o2 


'32 


33 


.3910 


.4081 


• 4252 


• 4422 


• 45 9 3 


•4762 


.4932 


•5ioo .5269 


•5437 


• 56o5 


33 


34 


.3913 


.4084 


• 4255 


• 4425 


•45 9 5 


.4765 


•4934 


•5io3 .5272 


• 544o 


• 56o8 


34 


35 


.3916 


.4087 


• 4258 


.4428 


.4598 


.4768 


•4937 


• 5io6 ; 0275 


• 5443 


• 56u 


35 


36 


.3919 


.4090 


.4261 


•443i 


.4601 


•477i 


.4940 


•5iu9 ! -5277 | 


.5446 


• 56i3 


36 


3 7 


.3922 


.4093 


• 4263 


•44M 


.4604 


•4773 


• 4 9 43 


•5lI2 


.5280 


.5448 


• 56i6 


37 


38 


• 3 9 25 


• 4096 


.4266 


•443 7 


.4607 


.4776 


•4946 


• 5i i5 


• 5283 


• 545i 


.5619 


38 


39 


.3927 


.4098 


.4269 


•443 9 


.4609 


•4779 


.4948 


• 5ii7 


.5286 


• 54M 


.5622 


3 9 


4o 
4 1 


• 3930 
.3933 


• 4ioi 
•4ic4 


.4272 

•42 7 5 


•4442 
•4445 


.4612 


.4782 


•495i 


•5l20 


.5289 


• 5457 
.5460 


O02D 
.5627 


40 
4i 


• 46i5 


•4785 


•4954 


• 5i23 


c 
.5291 


42 


3 9 36 


.4107 


•4278 


• 4448 


.4618 


.4788 


•4957 


.5126 .5294 


0462 


• 563o 


42 


43 


3939 


.4110 


.4280 


• 445 1 


.4621 


•479° 


• 4960 


0129 


0297 


• 5465 


• 5633 


43 


44 


.3942 


• 4n3 


• 4283 


• 4454 


.4624 


•4793 


.4963 


• 5i3i 


• 53oo 


• D468 


• 5636 


44 


£ 


• 3 9 45 


.4116 


.4286 


• 4456 


.4626 


•4796 


.4965 


• 5i34 


.53o3 


•5471 


• 5638 


45 


46 


•3947 


.4118 


.4289 


•4459 


.4629 


•4799 


• 4968 


• 5i3 7 


.53o6 


•5474 


564 1 


46 


47 


.3950 


.4121 


.4292 


.4462 


• 4632 


• 4802 


•4971 


•5i4o 


• 53o8 


•i47t» 


•5644 


47 


48 


• 3 9 53 


.4124 


.4295 


• 4465 


• 4635 


• 48o5 


•4974 


• 5i43 


• 53n 


• 5479 


•5647 


48 


49 


3966 


.4127 


.4298 


• 4468 


• 4638 


.4807 


•4977 


• 5i45 


• 53i4 


• 5482 


oooo 


-J9 


5o 


• 3 9 5 9 


• 4x3o 


• 43oo 


.4471 


• 464i 


.4810 


•4979 


•5i48 


.5317 ; 


• 5485 


• 00D2 


5o 


5i 


.3962 


• 4i33 


• 43o3 


•4474 


• 4643 


• 48i3 


.4982 


• 5i5i 


• 5320 


.5488 


.5655 


5i 


5? 


.3965 


• 4i35 


• 43o6 


•4476 


• 4646 


.4816 


• 49^5 


• 5i54 


• 5322 


•5490 


• 5658 


52 


53 


.3967 


• 4i38 


.4309 


•4479 


•4649 


.4819 


.4988 


• 5i57 


• 5325 


04 9 3 


0661 


53 


54 


.3970 


•4i4i 


• 43i2 


.4482 


• 4652 


.4822 


•499 1 


.5i6o 


• 5328 


.5496 


0664 


5-4 


55 


.3973 


.4:44'-43i5 


• 4485 


• 4655 


.4824 


•4994 


.5162 


•533i , 


• 5499 


. DODO 


55 


56 


.3976 


•4i47 


•43i7 


• 4488 


• 4658 


.4827 


.4996 


• 5i65 


• 5334 


• 55o2 


0009 


56 


57 


.3979 


• 4i5o 


• 43io 


.4491 


.4660 


• 483o 


.4999 


• 5i68 


• 5336 


• 55o4 


.5672 


57 


58 


.3982 


• 4i53 


• 4323 


.4493 


• 4663 


• 4833 


• 5cX>2 


• 5i7i 


.5339 


• 55o 7 

• 55ic 


• 56-5 


5S 


59 


.3985 


• 4i55 


.4326 


.4496 


.4666 


• 4836 


• 5oo5 


•5i74 -5342 


0678 


60 


.3987 


• 4i58 


.4329 


.4499 


.4669 


• 4838 


♦ 5oo8 


•5i 761-5345 1 


5di3 


• 56So 



10 





TABLE OF CHORDS: [Radius = 1.0000]. 


M. 


33° 

. 568o 


34° 

.5847 


35° 

.6014 


36° 

.6180 


37° 

.6346 


38° 
.65n 


39° 


40° 


41° 

.7004 


42° 
•7167 


43° 
• 733o 





.6676 


.6840 


i 


.5683 


• 585o 


.6017 


• 6i83 


.6349 


.65i4 


.6679 


.6843 


7007 


.7170 


. 7 333 


I 


2 


-5686 


.5853 


.6020 


.6186 


.6352 


.65i 7 


.6682 


.6846 


7010 


7i73 


• 7 335 


2 


3 


• 568 9 


.5856 


.6022 


.6189 


.6354 


.652o 


.6684 


.6849 


• 70T2 


.7 [76 


• 7 338 


3 


4 


.56 9 i 


.5859 


.6025 


• 6191 


• 635 7 


.6522 


.6687 


.685i 


•70l5 


7 r 7 8 


. 7 34i 


4 


5 


• 56 9 4 


• 586 1 


.6028 


.6194 


• 636o 


.6525 


.6690 


.6354 


• 7018 


• 7c8i 


•7344 


5 


6 


.56 97 


• 5864 


• 6o3i 


.6197 


.6363 


.6528 


.66 9 3 


.6857 


.7020 


•7>84 


• 7 346 


6 


7 


.5700 


• 5867 


• 6o34 


.6200 


.6365 


.653 1 


• 66 9 5 


.6860 


• 7023 


.7186 


.7349 


7 


8 


• 57o3 


• 58 7 o 


• 6o36 


.6202 


• 6368 


.6533 


.6698 


.6862 


.7026 


.7189 


• 7352 





9 


• 57o5 


• 58 7 2 


• 6039 


• 6205 


• 63 7 i 


.6536 


• 6701 


.6865 


.7029 


.7192 


•7354 


9 


IO 

ii 


.5708 
.5 7 n 


• 58 7 5 

• 58 7 8 


.6042 
• 6o45 


.6208 
.6211 


• 63 7 4 

• 63 7 6 


• 6539 


.6704 
.6706 


• 6868 
.6870 


• 7o3 I 
•7034 


.7195 


• 7 35 7 

• 736o 


10 
11 


• 6542 


.7197 


12 


•5 7 i4 


.588 1 


• 6o47 


.6214 


.63 79 


.6544 


.6709 


• 68 7 3 


•7037 


.7200 


.7362 


12 


i3 


.5717 


• 5884 


• 6o5o 


.6216 


.6382 


•6547 


.6712 


.6876 


.7040 


«72o3 


• 7 365 


i3 


i4 


.5719 


• 5886 


• 6o53 


.6219 


.6385 


• 655o 


• 6 7 i5 


.6879 


.7042 


.7205 


• 7 368 


i4 


i5 


.5722 


• 588 9 


• 6o56 


• 6222 


• 638 7 


.6553 


.6717 


.6881 


.7045 


.7208 


.7371 


i5 


16 


• 5 7 25 


• 58 9 2 


• 6o58 


.6225 


.6390 


• 6555 


.6720 


• 6884 


.7048 


.7211 


• 7 3 7 3 


16 


17 


.5728 


•58 9 5 


• 6061 


.6227 


.63 9 3 


• 6558 


• 6723 


.6887 


•7o5o 


.7214 


•7376 


17 


18 


• 5 7 3o 


•58 97 


• 6o64 


• 623o 


• 63 9 6 


• 656 1 


.6725 


.6890 


• 7o53 


.7216 


.7379 


18 


*9 


• 5 7 33 


•5900 


• 6067 


• 6233 


• 63 9 8 


• 6564 


.6728 


.6892 


• 7o56 


.7219 


• 7 38i 


l 9 


20 
21 


• 5736 
.5 7 3 9 


•5903 
• 5906 


• 6070 


• 6236 

• 6238 


• 64oi 

• 64o4 


• 6566 
.6569 


.6731 


• 68 9 5 


•7059 
.7061 


• 7222 
.7224 


• 7 384 


20 
21 


.6072 


•6 7 34 


.6898 


. 7 38 7 


22 


.5742 


.5909 


.6o 7 5 


.6241 


.6407 


• 65 7 2 


• 6 7 36 


.6901 


.7064 


.7227 


.7390 


22 


23 


0744 


.591 1 


.6078 


.6244 


• 6410 


• 65 7 5 


.6739 


.6903 


.7067 


.7230 


.73-92 


23 


24 


.5747 


•5 9 i4 


.6081 


•6247 


• 64 1 2 


• 65 77 


.6742 


.6906 


.7069 


.7232 


• 7 3 9 5 


24 


25 


.575o 


.5917 


• 6o83 


• 6249 


• 64i5 


• 653o 


•6 7 45 


. 6909 


.7072 


.7235 


- 7 3 9 8 


25 


26 


• 5 7 53 


0920 


.6086 


.6252 


• 64i8 


• 6583 


•6 7 47 


• 691 1 


.7075 


• 7238 


.7400 


26 


27 


• 5 7 56 


.5922 


.6089 


• 6255 


• 6421 


• 6586 


•6750 


.6914 


.7078 


.7241 


. 7 4o3 


27 


28 


• 5 7 58 


• 5925 


• 6092 


• 6258 


• 6423 


• 6588 


• 6 7 53 


.6917 


• 7080 


•7243 


.7406 


28 


29 


• 5 7 6i 


.5928 


• 6095 


• 6260 


• 6426 


• 65 9 i 


•6 7 56 


• 6920 


• 7 o83 


.7246 


.7408 


29 


3o 
3i 


• 5 7 64 


• 5 9 3 1 


.6097 


• 6263 
.6266 


• 6429 


• 65 9 4 
.6597 


•675* 
.6761 


6922 
• 6925 


.7086 
.7089 


.7249 
• 725i 


.7411 
•74i4 


3o 
3i 


.5767 


• 5 9 34 


.6100 


.6432 


32 


.5769 


.5936 


• 6io3 


• 6269 


.6434 


• 6599 


.6764 


.6928 


.7091 


.7254 


.7417 


32 


33 


.5772 


.5 9 3 9 


.6106 


.6272 


• 643 7 


• 6602 


.6767 


.6931 


.7094 


.7257 


.7419 


33 


34 


•5775 


.5942 


.6108 


.6274 


• 644o 


.66o5 


.6769 


• 6 9 33 


.7097 


.7260 


.7422 


34 


35 


•5778 


• 5 9 45 


• 6m 


.6277 


.6443 


.6608 


•6772 


• 6 9 36 


.7099 


.7262 


•7425 


35 


36 


.5781 


.5947 


• 6u4 


.6280 


• 6445 


.6610 


•6775 


.6939 


.7102 


• 7265 


.7427 


36 


37 


• 5 7 83 


• 5950 


• 6117 


• 6283 


• 6448 


.661 3 


.6777 


• 6941 


• 7io5 


.7268 


• 743o 


3 7 


38 


.5786 


• 5 9 53 


.6119 


.6285 


• 645 1 


.6616 


.6780 


•6 9 44- 


.7108 


.7270 


• 7433 


38 


3 9 


•5789 


.5956 


• 6122 


.6288 


.6454 


• 6619 


♦ 6 7 83 


•6947 


.7110 


.7273 


•7435 


39 


4o 
4i 


.5792 
• 5 79 5 


.5 9 5 9 


.6125 


• 6291 


• 6456 


• 6621 


.6786 
.6788 


• 6950 


• 7ii3 


.7276 
.7279 


• 7 438 

• 744i 


4o 
4i 


• 5961 


.6128 


.6294 


• 645 9 


.6624 


• 6952 


.7116 


42 


.5797 


• 5 9 64 


• 6i3o 


.6296 


.6462 


.6627 


.6791 


.6955 


.7118 


.7281 


•7443 


42 


43 


• 58oo 


.5967 


• 6i33 


.6299 


.6465 


• 663o 


•6794 


.6958 


.7121 


.7284 


•7446 


43 


44 


• 58o3 


.5970 


• 6i36 


• 63o2 


.6467 


• 6632 


.6797 


.6961 


.7124 


.7287 


•7449 


44 


45 


• 58o6 


.5972 


.6139 


.63o5 


.6470 


• 6635 


.6799 


.6963 


.7127 


.7289 


• 7452 


45 


46 


• 58o8 


.5975 


.6142 


• 63o 7 


• 64 7 3 


• 6638 


.6802 


.6966 


.7129 


.7292 


•7454 


46 


47 


• 58n 


•5978 


• 6i44 


• 63io 


.6476 


.6640 


• 68o5 


• 6969 


.7132 


.7295 


.7457 


47 


48 


• 58i4 


• 5 9 8i 


•6147 


• 63i3 


.6478 


• 6643 


.6808 


.6971 


• 7 i35 


.7298 


.7460 


48 


49 


• 58i 7 


• 5 9 84 


• 61 5o 


• 63i6 


• 648i 


• 6646 


.6810 


•6974 


.7137 


• 73oo 


.7462 


4o 


5o 
5i 


• 582- 
.5822 


.5986 
.5989 


• 6i53 

• 6i55 


• 63i8 
.6321 


• 6484 


.6649 


.681 3 
.6816 


.6977 


.7140 
•7i43 


• 73o3 


• 7 465 
.7468 


5o 
5i 


.6487 


• 665 1 


.6980 


• 73o6 


52 


• 58 2 5 


.5992 


• 6i58 


.6324 


.6489 


• 6654 


.6819 


.6982 


•7i46 


• 73o8 


-7471 


52 


53 


• 5828 


• 5 99 5 


.6161 


• 632 7 


• 6492 


• 665 7 


.6821 


.6985 


•7i48 


• 73 1 1 


.7473 


53 


54 


• 583 1 


.5997 


.6164 


.633o 


.6495 


.6660 


.6824 


.6988 


• 7i5 1 


•7^4 


•7476 


54 


55 


•5834 


.6000 


• 6166 


.6332 


.6498 


.6662 


.6827 


.6991 


•7i54 


• 73i6 


•7479 


55 


56 


•5836 


• 6oo3 


• 6169 


.6335 


• 65oo 


• 6665 


.6829 


.6993 


• 7r56 


.7319 


.7481 


56 


5 7 


• 583 9 


.6006 


• 6172 


• 6338 


• 65o3 


• 6668 


• 6832 


• 6996 


.7159 


.7322 


•7484 


5 7 


58 


• 5842 


• 6009 


.6175 


• 634i 


• 65o6 


.6671 


.6835 


.6999 


• 7162 


• 7 3 2 5 


•7487 


58 


59 


•5845 


• 601 1 


.6178 


.6343 


• 6509 


• 66 7 3 


• 6838 


• 7001 


• 7 i65 


.7327 


.7489 


59 


60 


•584 7 


• 6014 


.6180 


.6346 


• 65n 


.6676 


.6840 


.7004 


.7167 


• 733o 


.7492 


60 



11 



TABLE OF CHORDS: [Radius = 1.0000]. 


if. 

o' 


440 


45° 


46° 

. 7 8i5 


470 

.7975 


48° 
• 8i35 


49° 


50° 


51° 

.8610 


52° 
■8767 


53° 

• 8024 


54° 

• 9080 


M. 

O' 


.7492 


•7654 


.8094 


.8452 


i 


•74 9 ^> 


• 7 656 


.7817 


.7978 


• 8i3 7 


.8297 


.8455 


• 86i3 


•8770 


.8927 


.9082 


I 


2 


.7498 


.7659 


.7820 


.7980 


• 8i4o 


.8299 


.8458 


• 86i5 


.8773 


.8929 


.9085 


2 


3 


• 75oo 


.7662 


.7823 


. 79 83 


• 8i43 


.83o2 


.8460 


.8618 


.8775 


.8932 


.9088 


3 


4 


• 75o3 


.7664 


.7825 


.7986 


.8i45 


.83o4 


.8463 


.8621 


•8778 


.8934 


• 9090 


4 


b 


• 75o6 


.7667 


.7828 


.7988 


.8148 


.83o 7 


.8466 


• 8623 


.8780 


.8 9 3 7 


.9093 


5 


6 


• 75o8 


.7670 


. 7 83 1 


.7991 


• 8i5i 


• 83io 


.8468 


.8626 


• 8 7 83 


.8940 


.9095 


6 


*7 


• 75n 


.7672 


. 7 833 


•7994 


• 8i53 


• 83i2 


.8471 


.8629 


.8786 


.8942 


• 9098 


7 


8 


•75i4 


.7675 


. 7 836 


.7996 


• 8i56 


• 8?i5 


• 84 7 3 


.863 1 


.8788 


• 8 9 45 


.9101 


8 


9 


• 7 5i6 


.7678 


• 7 83 9 


•7999 


• 8i5 9 


• 83i8 


.8476 


• 8634 


.8791 


•8947 


• 9103 


9 


IO 

ii 


.7519 
.7522 


.7681 

• 7 683 


•784i 

•7844 


.8002 
.8004 


.8161 
.8164 


• 832o 


•8479 
• 848 1 


• 8636 


•8794 


• 8950 

• 8 9 53 


• 9106 


10 
11 


.8323 


• 863 9 


.8796 


.9108 


12 


.7524 


.7686 


.7847 


.8007 


.8167 


• 8326 


• 8484 


.8642 


.8799 


• 8 9 55 


• 9111 


12 


i3 


.7527 


.7689 


•7849 .8010 


.8169 


• 8328 


.8487 


.8644 


.8801 


.8958 


• 91 r3 


i3 


i4 


• 753o 


• 7691 


• 7 85 2 


.8012 


• 8172 


• 833i 


.8489 


.8647 


.8804 


.8960 


• 9116 


i4 


i5 


• 7 533 


.7694 


• 7 855 


• 8oi5 


• 8i 7 5 


.8334 


.8492 


.865o 


.8807 


• 8 9 63 


.9119 


i5 


t6 


• 7 535 


.7697 


. 7 85 7 


.8018 


•8177 


.8336 


.8495 


.8652 


.8809 


.8966 


.9121 


16 


17 


• 7 538 


•7699 


.7860 


'8020 


• 8180 


• 8339 


•8497 


• 8655 


.8812 


.8968 


.9124 


17 


18 


•754i 


.7702 


. 7 863 


.8023 


•8i83 


• 834i 


• 85oo 


• 865 7 


.8814 


•8971 


.9126 


18 


!9 


•7543 


• 77o5 


• 7 865 .8026 


• 8i85 


• 8344 


• 85o2 


.8660 


.8817 


.8973 


.9129 


19 


20 
21 


• 7 546 
.7549 


.7707 
.7710 


•7868 
.7871 


.8028 
• 8o3i 


•8188 
• 8190 


•8347 
.834 9 


• 85o5 

• 85o8 


.8663 


.8820 
.8822 


.8976 


.9132 


20 
21 


• 8665 


.8979 


.9134 


22 


. 7 55i 


• 7 7 i3 


.7873 


• 8o34 


• 8i 9 3 


.8352 


• 85io 


.8668 


.8825 


.8981 


•9 l3 7 


22 


23 


•7554 


• 77i5 


.7876 


• 8o36 


.8196 


.8355 


• 85i3 


.8671 


.8828 


.8984 


• 9 i3 9 


23 


24 


•755 7 


•77i8 


.7879 


.8039 


.8198 


• 835 7 


• 85i6 


• 86 7 3 


• 883o 


.8986 


.9142 


24 


25 


• 756o 


.7721 


.7882 


.8042 


.8201 


.836o 


• 85i8 


.8676 


• 8833 


.8989 


•9i45 


25 


26 


• 7 562 


•7723 


• 7 884 


• 8o44 


.8204 


.8363 


.8521 


.8678 


• 8835 


.8992 


.9147 


26 


27 


• 7 565 


.7726 


.7887 


.8047 


.8206 


.8365 


.8523 


.8681 


• 8838 


.8994 


• 900 


27 


28 


• 7 568 


.7729 


.7890 


• 8o5o 


• 8209 


.8368 


• 8526 


■ -8684 


• 884i 


.8997 


• 91D2 


28 


29 


.7570 


•773 1 


.7892 


• 8o52 


.8212 


• 83 7 i 


• 852 9 


•8686 


• 8843 


.8999 


• 9 i55 j 


29* 


3o 
3i 


. 7 5 7 3 
.7576 


.7734 
•7737 


• 7893 


• 8o55 


.8214 
.8217 


• 83 7 3 
.83 7 6 


• 853 1 

• 8534 


•8689 
.8692 


• 8846 


• 9002 


•9 l5 7 j 
.9160 


3o 
3i 


.7898 


• 8o58 


• 8848 


• 9005 


32 


. 7 5 7 8 


•774© 


.7900 


.8060 


.8220 


• 83 7 8 


• 853 7 


.8694 


.885 1 


.9007 


.9163 


32 


33 


• 7 58i 


•7742 


.7903 


• 8o63 


.8222 


.838i 


.853 9 


.8697 


.8854 


.9010 


• 9 i65 ; 


33 


34 


•7584 


•7745 


.7906 


.8066 


.8225 


.8384 


.8542 


.8699 


.8856 


.9012 


.9168 


34 


35 


. 7 586 


•7748 


.7908 


.8068 


.8228 


.8386 


.8545 


.8702 


.8859 


• 9015 


.9170 ; 


35 


36 


•758 9 


.7730 


.791 1 


.8071 


• 823o 


• 838 9 


.8547 


• 8 7 o5 


.8861 


.9018 


.9173 


36 


37 


.7592 


•7753 


.7914 


.8074 


• 8233 


.8392 


.855o 


.8707 


• 8864 


• 9020 


• 9176 


37 


38 


. 7 5 9 5 


•7756 -7916 


.8076 


• 8236 


• 83 9 4 


.8552 


.8710 


.8867 


■ 9023 


.9178 


38 


3 9 


.7597 


•7758 


.7919 


.8079 


• 8238 


• 83 97 


• 8555 


.8712 


.8869 


• 9020 


.9181 , 


3 9 


4o 
4i 


• 7000 


.7761 


.7922 


.8082 


.8241 

.8244 


• 84oo 


• 8558 


• 8 7 i5 


.8872 
.8874 


.9028 
.903 1 


• 9 .83 
.9186 


4o 
4i 


• 7 6o3 


• 7764 


.7924 


.8084 


.8402 


• 856o 


.8718 


42 


• 76o5 


.7706 .7927 


.8087 


.8246 


• 84o5 


.8563 


.8720 


•8877 


• 9033 


.9188 


42 


43 


.7608 


•7769 .7930 


• 8090 


.8249 


• 84o8 


.8566 


.8723 


.8880 


• 9036 


.9191 


43 


44 


• 7611 


.7772 


.7932 


• 8092 


.825 1 


• 84io 


.8568 


.8726 


.8882 


.9038 


.9194 


44 


45 


.7613 


•7774 


.7935 


• 8095 


.8254 


• 84i3 


• 85 7 i 


.8728 


• 8885 


.9041 


•919° 


45 


46 


.7616 


•7777 


. 79 38 


.8098 


.8257 


• 84i5 


• 85 7 3 


• 8 7 3 1 


.8887 


.9044 


•9 I 99 


46 


47 


.7619 


7780 


.7940 


•8100 


.82D9 


.8418 


.8576 


•8 7 34 


.8890 


• 9046 


• 9201 


47 


48 


• 7621 


.7782 


.7943 


•8io3 


.8262 


.8421 


• 85 79 


.8736 


.8893 


•9049 


.9204 


48 


49 


,6a4 


• 7785 


•7946 


•8io5 


• 8265 


• 8423 


• 858i 


•8739 


.8895 


• 905 1 


.9207 ; 


49 


5o 
5i 


.7627 
.7629 


.7788 
•779 1 


.7948 
.795 1 


•8108 


.8267 


.8426 
.8429 


• 8584 
•858 7 ! 


•8 7 4i 
.8744 


• 8898 


• 9054 


•9209 

1 


So 
5i 


.8111 


.8270 


• 8900 


.9056 


•9212 


5a 


• 7 632 


.7793 


.7954 


• 8n3 


.8273 


.843 1 


.8589 


8747 -8903 


• 9 od 9 


.9214 


52 


53 


. 7 635 


.7796 


.7936 


.8116 


.8275 


.8434 


.8592 


.8749 


.8906 


.9062 


.9217 


53 


54 


• 7 638 


•7799 


•79:59 


81 19 .8278 


.843^ 


•85 9 4 


.8752 


.8908 


• 9064 


. 9 2[ 9 


5^ 


55 


.7640 


.7801 


79^2 


•8121 .8281 


• 843 V 


• 8b 97 


•8 7 54 


• 8911 


.9007 


•9222 


DD 


56 


.7643 


.7804 


•79 6 4 


.8124-8283 


.8442 


• 8600 


• 8 7 5 7 


.8914 


• 9069 


•922D 


56 


57 


.7646 


•7807 


.7967 


.8127I.8286 


•8444 


.8602 


.8760 


.8916 


.9072 


.9227; 


5? 


58 


• 7 648 


.7809 


.7970 


.8129 .8289 


•8447 


• 86o5 


.8762 


• 8919 


.9070 


.923o 


58 


59 


765 1 


.7812 


.7972 


•8i32 .8291 


• 845o 


• 86oS 


.8765 


.8921 


.9077 


•9232 


60 


• 7 654 


• 7 8£5 


.7975 


.8i35|-8294 


.8452 


.8610 


•8767 


•89^4 


•9080. "Q235 



12 



TABLE OF CHORDS 


; [Radius = 1.0000]. 




m. 

o' 


55° 

.9235 


56° 

.9389 


57° 
.9543 


68° 
.9696 


59° 


6O 


61° 


62° 


63° 


64° 


M. 

O' 


.9848 


1 .0000 


1 -ot5i 


1 .o3oi 


i*o45o 


1.0598 


i 


.9238 


.9392 


• 9 546 


.9699 


• 9 85 1 


1 .ooo3 


i.oi53 


1 «o3o3 


i*o45a 


I -cOol 


I 


2 


.9240 


.9395 


• 9 548 


.9701 


. 9 854 


1 .ooo5 


i.oi56 


1 «o3o6 


i*o455 


1 • o6o3 


2 


3 


.9243 


.9397 


• 9 55i 


.9704 


• 9 856 


1 . 0008 


i.oi58 


i.o3o8 


i*o457 


1 - 0606 


3 


4 


.9245 


.9400 


.9553 


.9706 


• 9 85 9 


1 .0010 


1 .0161 


1 • o3 1 1 


1 • 0460 


1-0608 


4 


5 


.9248 


.9402 


* 9 556 


.9709 


.9861 


1 .ooi3 


1 «oi63 


1 .o3i3 


1 «o462 


1 -0611 


5 


6 


.9250 


• 94o5 


• 9 55 9 


.9711 


.9864 


1 .ooi5 


1 • 1 66 


1 «o3i6 


i-o465 


1 «o6i3 


6 


7 


.9253 


.9407 


* 9 56i 


.9714 


.9866 


1. 0018 


1. 0168 


i.o3i8 


1-0467 


1 *o6i6 


7 


8 


.9256 


.9410 


• 9 564 


•97i7 


.9869 


1.0020 


1.0171 


1 -o32i 


1-0470 


1*0618 


8 


9 


.9258 


.941 3 


• 9 566 


.9719 


.9871 


1. ooa3 


1 -0173 


1 «o323 


1.0472 


1 .0621 


9 


IO 

ii 


• 9261 
.9263 


.941 5 
.9418 


• 9569 
.9571 


.9722 
•9724 


•9»74 
.9876 


1 .0025 


1-0176 


1 -o326 


i* o475 


1*0623 


10 
11 


1 .0028 


1. 0178 


i.o328 


1.0477 


1 .0626 


12 


.9266 


.9420 


• 9574 


.9727 


.9879 


i.oo3o 


1.0181 


i*o33i 


1.0480 


1.0628 


12 


i3 


.9268 


.9423 


.9576 


.9729 


.9881 


i«oo33 


i.oi83 


i-o333 


1.0482 


1 *o63o 


i3 


U 


.9271 


• 9425 


.9579 


.9732 


.9884 


i.oo35 


1.C186 


i.o336 


i-o485 


i*o633 


i4 


i5 


.9274 


.9428 


* 9 58i 


.9734 


.9886 


i.oo38 


1. 0188 


i.o338 


1.0487 


i*o635 


i5 


16 


.9276 


• 943o 


• 9 584 


.9737 


.9889 


1 • oo4o 


1.0191 


i«o34i 


1.0490 


i*o638 


16 


17 


.9279 


• 9433 


. 9 58 7 


•9739 


.9891 


1 .oo43 


1 .0193 


i*o343 


1 .0492 


1 • 0640 


17 


18 


.9281 


• 9436 


• 9 58 9 


.9742 


.9894 


1 «oo45 


1 • 1 96 


i*o346 


1*0495 


1 *o643 


18 


19 


.9284 


• 9 438 


. 9 5 9 2 


•9744 


.9897 


1.0048 


1*0198 


i*o348 


1.0497 


i*o645 


*9 


20 
21 


.9287 


•944i 
• 9443 


•9^94 


•9747 


.9899 
.9902 


1 «oo5o 


I-020I 


i*o35i 


1 -o5oo 


1.0648 


20 
21 


.9289 


.9597 


.9750 


1 -0053 


I«0203 


i-o353 


1 «o5o2 


1 .o65o 


22 


.9292 


•9446 


•9 5 99 


.9752 


.9904 


i.oo55 


I-0206 


i-o356 


1 -o5o4 


i.o653 


22 


23 


.9294 


•9448 


• 9602 


.9755 


.9907 


i.oo58 


I -0208 


i.o358 


1 .0507 


i.o655 


23 


24 


.9297 


•945 1 


• 9604 


.9757 


.9909 


1 • 0060 


I .0211 


1 -o36i 


i«o5o9 


i.o658 


24 


25 


.9299 


•9454 


• 9607 


.9760 


.9912 


1 • oo63 


I «02l3 


i.o363 


1 • o5 1 2 


1 • 0660 


25 


26 


.9302 


• 9 456 


• 9610 


.9762 


.9914 


1 -oo65 


I .0216 


i.o366 


i*o5i4 


1 -0662 


26 


27 


«93o5 


•9459 


• 9612 


. 97 65 


.9917 


1.0068 


I. 0218 


i.o368 


i*o5i7 


i.o665 


27 


28 


.9307 


• 9461 


• 9615 


.9767 


.99.9 


1 .0070 


1*0221 


1 .0370 


i*o5i9 


1 .0667 


28 


29 


• 9310 


•9464 


• 9617 


.9770 


.9922 


1 -0073 


I«0223 


1.0373 


1 *o522 


1 .0670 


29 


3o 
3i 


• 9312 


.9466 


• 9620 


.9772 

•9775 


.9924 
.9927 


1.0075 


1.0226 


i.o3 7 5 


i*o524 


1*0672 


3o 
3i 


• 93i5 


•9469 


.9622 


1 • 0078 


I.0228 


i.o3 7 8 


1*0527 


1.0675 


32 


•9 3l 7 


.9472 


.9625 


•9778 


.9929 


1 .0080 


I .023l 


i.o38o 


1 *o529 


1 * 0077 


32 


33 


• 9320 


•9474 


.9627 


.9780 


.9932 


i.oo83 


i .0233 


i-o383 


i*o532 


1.0680 


33 


34 


. 9 3 2 3 


•9477 


.9630 


. 97 83 


•9934 


1.0086 


r -0236 


i.o385 


i*o534 


1.0682 


34 


35 


. 9 3 2 5 


•9479 


• 9 633 


9~85 


.9937 


1.0088 


L0238 


i.o388 


1 *o537 


i.o685 


35 


36 


.9328 


.9482 


.9635 


9788 


.9939 


1 .0091 


1. 0241 


r .0390 


i*o539 


1.0687 


36 


37 


.9330 


•9484 


• 9638 


.9790 


.9942 


1.0093 


1.0243 


1 .0393 


i*o542 


1 .0690 


3 7 


38 


.9333 


•9487 


• 9640 


.9793 


.9945 


1 • 0096 


1 .0246 


1 .0395 


i*o544 


1*0692 


38 


3 9 


.9335 


•9489 


.964? 


.9795 


•9947 


1 .0098 


1.0248 


1.0398 


i*o547 


1 • 0694 


39 


4o 
4i 


• 9338 


.9492 
•9495 


.9645 


.9798 
• 9800 


• 9960 
.9952 


LOIOI 


I'025l 


1 «o4oo 


1*0549 


1.0697 


4o 
4i 


.9341 


.9648 


1 »oio3 


I. 0253 


i*o4o3 


i*o55i 


1*0699 


42 


.9343 


•9497 


.9650 


.9803 


.9955 


1 .0106 


1.0256 


i*o4o5 


i*o554 


1*0702 


42 


43 


.9346 


• 9500 


• 9 653 


.9805 


.9957 


1*0108 


I.0258 


1.0408 


i*o556 


1 .0704 


43 


44 


.9348 


• 9502 


.9655 


.9808 


.9960 


I.OIII 


1.0261 


1 *o4io 


1*0559 


1 .0707 


44 


45 


.9351 


• 95o5 


• 9 658 


• 0810 


.9962 


1 «oi i3 


1 .0263 


i. 04 1 3 


1 *o56i 


1-0709 


45 


46 


.9353 


• 9.507 


.9661 


• 9 8i3 


• 996b 


1 .0116 


1.0266 


i*o4i5 


i*o564 


1 -0712 


46 


47 


• 9 356 


• 9510 


• 9 663 


.9816 


.9967 


1.0118 


1.0268 


1.0418 


i*o566 


1*0714 


47 


48 


• 9 35 9 


• 9512 


.9666 


.9818 


•9970 


1.0121 


1*0271 


i* 0420 


1 .0569 


1*0717 


48 


49 


• 9361 


•95i5 


• 9668 


.9821 


.9972 


1 .0123 


1*0273 


i*o423 


i« 0571 


1*0719 


49 


5o 
5i 


• 9364 
. 9 366 


• 9 5i8 


.9671 
.9673 


.9823 


• 997b 
•9977 


1. 0126 


1.0276 


i* 0425 


i* 0574 


1*0721 


5o 
5i 


■ 9520 


.9826 


1. 0128 


1.0278 


1.0428 


1.0576 


1.0724 


52 


.9369 


• 9 523 


.9676 


.9828 


.9980 


1 .oi3i 


1. 0281 


1 *o43o 


1.0579 


1 ,.0726 


52 


53 


.9371 


• 9525 


.9078 


• 9 83 1 


.9982 


i*oi33 


1.0283 


i*o433 


i*o58i 


1.0729 


53 


54 


• 9 3 7 4 


.9528 


.9681 


.9833 


• 99 85 


1 >oi36 


1.0286 


i.o435 


i*o584 


1.073 1 


54 


55 


•9 3 77 


• 953o 


.9683 


• 9836 


.9987 


i-oi38 


1.0288 


i-o438 


i*o586 


1.0734 


55 


56 


.9379 


.9533 


.9686 


• 9 838 


.9990 


1 .oi4i 


1 .0291 


1 »o44o 


i*o58 9 


1.0736 


56 


57 


. 9 382 


• 9 536 


.9689 


.9841 


.9992 


i»oi43 


1*0293 


i.o443 


1*0591 


1.0739 


5 7 


58 


• 9384 


• 9 538 


.9091 


.9843 


.9995 


i«oi46 


1*0296 


I-0445 


1*0593 


1. 0741 


58 


5 9 


. 9 38 7 


•954i 


.9694 


.9846 


.9998 


1.0148 


1 .0298 


1.0447 


1*0596 


1*0744 


59 


60 


• 9 38 9 


.9543 


•9696 


.9848 


1 0000 


loi5i 


1 *o3oi 


i«o45o 


1.0598 


1.0746 


00 



13 







TABLE OF CHORDS: [Ra 


DIUS = l.OOOu]. 




0' 


65° 


66° 


67° 


6§° 


69° 


70° 


| 71° 


72° 


73° 


O 


1.0746 


1-0893 


1. 1039 


I1.1184 


1 .1328 


1. 1472 


i- 1614 


1 -1756 


1. 1896 


I 


1.0748 


1.0895 


i«io4i 


1-1186 


i.i33i 


1. i474 


1.1616 


1. 1758 1. 1899 


I 


a 


I-07DI 


1-0698 


1. io44 


1.7189 


i.i333 


1.1476 


; 1.1619 


1. 1760 


1. 1901 


2 


3 


1.0753 


1 . 0900 


1. 1046 


1 . 1191 


1.4335 


1. 1479 


' 1.1621 


i.i 7 63 


1 .1903 


3 


4 


1.0756 


1.0903 


1. 1048 


1 . 1 1 9 4 


i.i338 


i.t48i 


i 1. 1624 


1-1765 


1 .1906 


4 


5 


i-o 7 58 


1 .0903 


i-io5i 


i- 1 196 


1. 1 34o 


i.i483 


! i- 1626 
! 1. 1628 


1. 1767 


1. 1908 


5 


6 


1-0761 


1-0907 


i.io53 


1 119S 


i.i342 


ii i486 


1-1770 


1 .1910 





7 


1 -0763 


1 .0910 


1 - io56 


1 • 1201 


1. 1 U3 


1.1488 


! i-i63i 


1-1772 


1 -1913 


" 


8 


1.0766 


1 .0912 


i-io58 


I • I2o3 


i-i34- 


1.1491 


i i-i633 


i-i775 


1 . 1913 


8 


9 


1.0768 


1 -0913 


1 ■ 1001 


I • 1206 


i-i35o 


1.1493 


i i-i635 


1 -1777 


1. 1917 


9 


IO 

ir 


1-0771 


1*0917 


i.io63 


I- I 208 


1 • i352 


1-1495 


\ i.i638 


I- 1779 


1-1920 


10 
ii 


1-0773 


1 .0920 


1 • io65 


I • I 2IO 


1 -i354 


1. 1498 


; 1.1640 


I. 1782 


1. 1922 


12 


1 -0775 


1 -0922 


1. 1068 


I - I2l3 


1 • i357 


I .1300 


! 1. 1642 


1. l 7 84 


1. 1924 


ii. 


i3 


1.0778 


1 .0924 


1 • 1070 


I - 1213 


1 • i359 


I • l5o2 


; 1. i645 


1. 1786 


1. 1927 


i3 


i4 


1.0780 


1 -0927 


1. io-3 


I-12I6 


i- 1 362 


I- 1303 


; 1. 1647 


1. 1789 


1 • 1929 


i4 


i5 


. 1.0783 


1 -0929 


1 • 1073 


1 - I220 


i-i364 


I .1307 


; li65o 


I.1791 


1 .1931 


i5 


16 


1.0785 


1 -0932 


1. 1078 


I - 1222 


i.i366 


i-i5io 


i.i652 


1. l 79 3 


1. 1934 


16 


17 


1.0788 


1-0934 


1.1080 


I • 1223 


i. 1369 


Ll5l2 


; 1.1654 


1. 1790 


1. 1 936 


1- 


18 


1-0790 


1.0937 


1. 1082 


I • 1227 


1 • 1371 


i-i5i4 


j i-i657 


1. 1798 


1. 1938 


iS 


19 


1-0793 


i.o 9 3 9 


I-I0&5 


I • I23o 


i-i3 7 4 


i.i5i 7 


1-1659 


I • 1800 


1-1941 


J 9 


20 
21 


1-0793 


1 -0942 


1. 1087 


I- 1232 


i.i3 7 6 


I.I3I 9 


1.1661 


i-i8o3 


1. 1943 


20 
21 


1.0797 


1 • 0944 


1 • 1090 


1-1234 


i.i3 7 8 


I • l522 


! 1. 1664 


i.i8o5 


1. 1946 


22 


1 • 0800 


1-0946 


1 • 1092 


1 • 1237 


li38i 


i-i524 


1. 1 666 


1 • 1 807 j 1 . 1 948 


22 


a3 


1.0802 


1-0949 


1.1094 


1 • 1239 


li383 


i-i526 


1. 1668 


1 -1810 


1 .1930 


23 


24 


i.o8o5 


1.093 1 


1. 1097 


1 .1242 


i.i3S6 


1-1529 


1.1671 


1. 1812 


1 -1932 


24 


25 


1.0807 


1.0954 


1.1099 


1. 1244 


i.i388 


1-153! 


. i-i6 7 3 


1.1814 


1. 1933 


25 


26 


i.o8ro 


1-0956 


1 « 1102 


1-1246 


1.1390 


i.i533 


1. 1676 


1. 1817 


1-1937 


26 


27 


1. 0812 


1.0939 


1 .1104 


1. 1 249 


i-i3 9 3 


i-i536 


1.10-8 


1.1819 


1. 1939 


2- 


28 


i-o8i5 


1.0961 


1. 1 107 


I « 1231 


1 • 1395 


i.i538 


1. 1680 


1 • 1821 


1. 1 962 


28 


29 


1. 0817 


1 • 0963 


1-1109 


I" 1254 


i.i3 9 8 


i-i54i 


i-i683 


1.1824 


1. 1 964 


29 


3o 
3r 


1-0820 


1 -0966 


1. mi 


I-I230 


1 -i4oo 


i-i543 


i-i685 


1.1826 


1. i960 


3o 
3i 


1-0822 


1.0968 


, III4 


I. 1258 


1 -i4o2 


1. i545 


1. 1687 


1. 1829 


1.1969 


32 


1.0824 


1. 0971 


1.1116 


I .1261 


r-i4o5 


i-i548 


i- 1690 


i.i83i 


1-1971 


32 


33 


1.0827 


1 -0973 


1.1119 


i- 1263 


1. 1407 


I- 1330 


1. 1092 


I-.I833 


1. 1973 


33 


34 


1 -0829 


1 -0976 


1 • 1121 


1 .1266 


1.1409 


I-I332 


! I. 1094 


i-i836 


1. 1976 


34 


35 


i-o832 


1.0978 


1-1123 


1. 1 268 


1 .1412 


1.1555 


! 1-1097 


1. 1838 


1.1978 


35 


36 


i-o834 


1-0980 


I -1120 


1.-1*71 


1.1414 


i-i55 7 


■ 1. 1099 


1.1840 


1. 1980 


36 


3 7 


i-o83 7 


1 -09^3 


I-II28 


1. 1273 


r-i4i7 


i-i56o 


1 .1702 


1-1843 


1. 1953 


3- 


38 


1.0839 


i.o 9 85 


i.ii3i 


1-1273 


1. 1419 


i-i562 


1. 1704 


1 . i845 


1. 1 953 


5? 


89 


1. 0841 


1.0988 


i-i-133 


1. 1278 


1.1421 


i-i564 


1.1706! 


1.1847 


1. 1987 


39 


4o 
4i 


1.0S44 


1 -0990 


i«n36 


1. 1280 


1.1424 


i-i567 


1 -1709. 


1. i85o 


1 .1990 
1 • 1992 


4o 
4i 


1.0846 


1.0993 


i-ii38 


1.1283 


1. 1426 


1.1569 


1-1711 


i.i852 


42 


1.0849 


1.0993 


1.1140 


1. 1 285 


1. 1429 


1. 1571 


i-i7i3 


i.i854 


1-1994 


42 


43 


i-o85i 


1.0997 


1.H43 


1 • 1257 


f.i43i 


I-I5-4 


1.1716' 


1.153- 


i-i997 


43 


44 


i-o854 


1 • 1000 


[-1145 


1.1290 


1U433 


1. 1576 


1.1718 


1.1809 


1.1999 


44 


45 


i-oS56 


1 • 1002 


1. 1 148 


1-1292 


i.?438 


1. 1379 


1 • 1 720 i 


1-I50I 


I -200I 


45 


46 


1-0859 


1. 1 003 


1 .n5o 


1 • 1295 


i-i58i 


1 « 1723 


1.1864 


1 .2004 


46 


47 


1. 0861 


1. 1007 


I - Il52 


1. 1 297 


1. 1 44 1 


i-i583 


1 .1725! 


1 • 1566 


I -2000 


47 


48 


i-o863 


1 • 1010 


i-ii55 


1 • 1 299 


I -1443 


i-i586 


1-17*7 


I- 1505 


1 -2008 


48 


49 


1-0866 


1 • 1012 


1.1157 


I • l302 


i-i445 


i-i588 


i.i 7 3o 


I-I5-I 


I -20II 


49 


5o 
5 1 


1-0868 


1 • 1014 


l • llOo 


i.i3o4 


1.1448 


1. 1 390 


i.i 7 32 


I.1873 


i-aoi3 ! 


53 

5i 


1. 0871 


1 • 1017 


I • 1 162 


i. i3o7 


i.i45o 


i.i5 9 3 


1. 1735 


. .1875 


I -20l5 


52 


1.0873 


1-1019 


I -1103 


1 • 1 309 


1. i452 


1-1393 


i-i-o- 




I -20lS 


52 


53 


1.0876 


I • 1022 


•.II67 


1 • i3i 1 


i. i455 


1-1398 


1.1739 


1 • 1880 


1-2020 


53 


54 


1.0878 


I - 1024 


I .1169 


i.i3i4 


i.i45 7 


1 • 1000 


1*1742 


: • ; rr_ 


I -2022 


54 


55 


1.0881 


I • IO27 


1 • II72 


i.i3i6 


1.1460 


i- 1602 


I -1-44 


1 « i885 


I -2G25 


55 


56 


i-o883 


I • IO29 


I.H74 


1 • i3 19 


1. 1462 


i-i6o5 


1-1746 




I-2027 


56 


57 


i-o885 


i-io3i 


I.II77 


1-1321 


1. 1 464 


1 -1007 


1-1-49 




I .2029 


57 


58 


1.0888 


i.io34 


I. I 179 


i-i323 


1 • *«t '' 


1 • 1609 


[-1-31 


1 • 1892' 1 «2032 


58 


5 9 


1.0890 


i«io36 


I.Il8l 


1.1326 


I. 1469' 


1 -.612 


I-I-53 


1 • i 894 1 • 2o34 


59 


S| 


1. o8 9 3 | 


1. 1039 


I-IIS4 


i.i3a8 


I-I472 


1. 1014 


I. I 7 DO ( 


1 • 1 500 1 • 2o36 i 60 



14 







TABLE OF CHORDS 


; [Radius = 1. 


OOOO j. 




M. 

o' 


740 


75° 


76° 


77° 


78° 


79° 


8O 


81° 


82° 


0' 


1 • 2o36 


i-2i 7 5 


i. 2 3i3 


1 .2450 


1.2586 


1 .2722 


1.2856 


1.2989 


I .3l2I 


I 


1 .2039 


1. 2178 


i.23i6 


1.2453 


1.2589 


1. .2724 


1.2858 


1.2991 


i.3i23 


1 


2 


I • 2o4 I 


f .2180 


i.23i8 


1.2455 


1 .2591 


1 .2726 


1.2860 


1.2993 


1. 3i26 


2 


3 


; I -2o43 


1. 2182 


1 .2320 


1.2457 


1.2593 


1 .2728 


1.2862 


1 • 2996 


1.3128 


3 


4 


I • 2o46 


1.2184 


I .2322 


1 . 2459 


1 .2595 


1 .2731 


1.2865 


1 • 2998 


1 -3i3o 


4 


5 


! I.2o4» 


1. 2187 


1.2325 


1 .2462 


1.2598 


1.2733 


1.2867 


1 «3ooo 


i.3i32 


5 


6 


j I • 2C)5o 


1. 2189 


I .3327 


1 . 2464 


1 . 2000 


1.2735 


1.2869 


I-3002 


i-3i34 


6 


7 


i i «2o53 


1 • 2191 


I.2.329 


1 . 2466 


1 .2602 


1 • 2737 


1 .2871 


1 • 3oo4 


i.3i37 


7 


8 


j i.2o55 


1. 2194 


1.2332 


1 . 2468 


1 • 2604 


1.2740 


1.2874 


1 .3007 


i-3i39 


8 


9 


1.2057 


1. 2196 


1.2334 


1-2471 


1 .2607 


1-2742 


1.2876 


1 • 3009 


i-3i4i 


9 


IO 

— 
ii 


1 • 2060 


1. 2198 


1.2336 


1-2473 


1 • 2609 


1.2744 


1.2878 


i-3oii 


i-3i43 


10 
11 


1 .2062 


1. 2201 


1.2338 


1.2475 


1 .261 1 


1 • 2746 


1.2880 


1 .3oi3 


i.3i45 


12 


1 • 2064 


I .2203 


1. a34i 


1 . 2478 


1 .2614 


1 • 2748 


1.2882 


1 «3oi5 


i.3i47 


12 


i3 


l 1 • 2066 


I • 22o5 


1.2343 


1.2480 


1-2616 


1 -2751 


1.2885 


i.3oi8 


i.3i5o 


i3 


i4 


1 • 2069 
1 .2071 


I .2208 


1.2345 


1.2482 


1.2618 


i.2 7 53 


1.2887 


1 .3020 


i.3i52 


i-4 


i5 


I «22lO 


1.2348 


1 . 2484 


1 .2620 


i.2 7 55 


1.2889 


I «3o22 


i-3i54 


i5 


16 


1 • 2073 


I .2212 


1 «235o 


1.2487 


1 .2623 


1.2757 


1.2891 


I -3o24 


1 . 3 1 56 


16 


n 


1 .2076 


I »22l4 


1.2352 


1 • 2489 


1 .2625 


1 . 2760 


1.2894 


1 .3027 


i-3i58 


17 


18 


1 • 2078 


1 .2217 


1.2354 


1. 2491 


1 .2627 


1.2762 


1.2896 


I .3029 


i.3i6i 


18 


J 9 


1 .2080 


I .2219 


1.2357 


1 . 2493 


1 -2629 


1 - 2764 


1 - 2898 


1 -3o3i 


i.3i63 


19 


20 
21 


1 • 2o83 


1 «222I 


1.2359 


1 • 2496 


I -2632 


1 .2766 


1 - 2900 


i.3o33 


i.3i65 


20 
21 


i.2o85 


1*2224 


i-236i 


1.2498 


1-2634 


1-2769 


1 • 2903 


1 «3o35 


1 . 3 1 67 


22 


1 . 2087 


1.2220 


1 • 2364 


1 -25oo 


1.2636 


1.2771 


1 - 2905 


i.3o38 


1 . 3 1 69 


22 


23 


1 • 2090 


1-2228 


1-2366 


i.25o3 


1.2638 


1.2773 


1 . 2907 


1 . 3o4o 


1 .3172 


23 


24 


1 • 2092 


I «223l 


1-2368 


1 • 25o5 


1 • 264 1 


1.2775 


1 .2909 


1 • 3o42 


i.3i 7 4 


24 


25 


1 • 2094 


1.2233 


1 .2370 


1 .2507 


1.2643 


1.2778 


1 -291 i 


i.3o44 


i.3i 7 6 


25 


26 


1 • 2097 


1-2235 


1.2373 


1-2509 


1.2645 


1 .2780 


1.2914 


i-3o46 


i.3i 7 8 


26 


27 


1 • 2099 


1.2237 


1.2375 


I -25l2 


1 . 2648 


1.2782 


1 .2916 


1 • 3o49 


i-3i8o 


27 


28 


1 .2101 


I .2240 


1.2377 


1 -25i4 


1 -265o 


1.2784 


1. 2918 


i.3o5i 


i-3i83 


28 


29 


1. 2104 


I .2242 


i-238o 


1 -25i6 


I -2652 


1.2787 


1 • 2920 


1 . 3o53 


i.3i85 


29 


3o 
3i 


1. 2106 


I .2244 


1-2382 


i-25i8 


1.2654 


1.2789 


1 .2922 


i.3o55 


1. 3187 


3o 
3i 


1.2108 


1.2247 


1.2384 


I - 2521 


1.2656 


1. 279 1 


1 .2925 


i.3o5 7 


1. 3189 


32 


1*2111 


1 .2249 


1.2386 


1,2523 


1 -2659 


1.2793 


1.2927 


1 • 3 060 


1 .3191 


32 


33 


I • 21 l3 


1.225l 


i.238 9 


I .2525 


1 .2661 


1.2795 


1 • 2929 


1 • 3o62 


1.3193 


33 


34 


I '21 l5 


1.2254 


1 .2391 


1-2528 


1 . 2663 


1.2798 


1 -2931 


1 • 3o64 


1 • 3 1 96 


34 


35; 


I '21 17 


I -2256 


1 .2393 


1 -253o 


1 . 2665 


1 • 2800 


1.2934 


1 . 3o66 


1.3198 


35 


36 


I .2120 


1.2258 


1 -2396 


1-2532 


1 . 2668 


1 .2802 


1 -2936 


i-3u68 


1 «32oo 


36 


3? 


1 .2122 


I .2260 


1.2398 


1-2534 


1 • 2670 


1 .2804 


1-2938 


1 .3071 


1 .8202 


3 7 


38 


I -2124 


I .2263 


1 • 2400 


i-253 7 


1 .2672 


1.2807 


1 • 2940 


1 .3073 


1 .3204 


38 


3 9 ! 


I-2I27 


1.2265 


1 .2402 


1 -2539 


1.2674 


1 .2809 


1 -2942 


1 -3o75 


1 .3207 


3 9 


.4o j 
4i 


I '2129 


I .2267 


1 • 24o5 


1. 2541 
1.2543 


1 -2677 


i- 2811 


1 • 2945 


1.3077 


1 .3209 


4o 
4i 


I >2l3l 


1.2270 


1 • 2407 


1 • 2679 


1.2813 


1 • 2947 


1.3079 


1 .3211 


42 


1.2134 


1 2272 


1 . 2409 


1.2546 


1. 2681 


1.2816 


1.2949 


i-3o82 


i.32i3 


42 


43 


i.2i36 


1.2274 


1 .2412 


1.2548 


1 • 2683 


1.2818 


1. 295 1 | 


i-3o84 


i.32i5 


4^ 


44 


i.ai38 


1.2277 


1.2414 


i-255o 


1 • 2686 


1 .2820 


1.2954 


i.3o86 


1. 3218 


44 


45 


1.2141 


1.2279 


1 .2416 


I «2D^2 


1 • 2688 


1 .2822 


1 -2956 


i.3o88 


1.3220 


45 


46 


1. 2i43 


I. 2281 


1.2418 


1-2555 


1 • 2690 


1.2825 


i.2 9 58 


1 . 3090 


I -3222 


46 


47 


1. 2i45 


1.2283 


1 .2421 


1.2557 


1 • 2692 


1.2827 


1 • 2960 


1 • 3093 


I .3224 


47 


48 


1.2148 


1 .2286 


1 .24a3 


1.2559 


1 • 2695 


1.2829 


1.2902 


1 .3095 


I -3226 


48 


49 


1 • 2 i 5o 


1.2288 


1.2425 


I -2562 


1 • 2697 


i..283i 


1 • 2965 


1.3097 


1.3228 


49 


5o 
5i 


I »2l52 

1.2154 


I .2290 


1.2428 


1 • 2564 


1 • 2699 


L2833 


1 • 2967 


1 • 3099 


i.323i 


5o 
5i 


I .2293 


i.243o 


1.2566 


1 .2701 


1.2836 


1 • 2969 


i.3ioi 


1.3233 


52 


1 .2157 


I .2295 


1 .2432 


1 • 2568 


1 • 2704 


1.2838 


1. 2971 


i«3io4 


1.3235 


52 


53 


1. 2 1 59 


1.2297 


1.2434 


1 -2571 


1 • 2706 


1 • 2840 


1.2973 


i-3io6 


i.323 7 


53 


54 


1.2161 


I .2299 


1.2437 


1.2573 


1 • 2708 


1.2842 


1.2976 


i.3io8 


i.3?3 9 


54 


55 


1. 2 164 


I«23o2 


1.2439 


1.2575 


1 .2710 


1.2845 


1.2978 


1 -3i 10 


1 .3242 


55 


56! 


1. 2 1 66 


1 • 23o4 


1.2441 


1.2577 


1 -2713 


1.2847 


1 .2980 


i»3i 12 


1.3244 


56 


5 7 


1. 2168 


I.2J06 


1-2443 


1 «258o 


1 -2715 


1.2849 


1 .2982 


i.3u5 


1.3246 


57 


58 


1 -2171 


1 .2309 


1 • 2446 


L2582 


1.2717 


i-285i 


1.2985 


1 .3117 


1.3248 


58 


59 


1 .2173 


I -23 1 I 


1.2448 


1 • 2584 


1-2719 


1.2854 


1.2987 


1.3119 


1 »325o 


59 


60 


1. 2 1 75 


i-23i3 


1 • 245o 


1-2586 


1.2722 


1-2.856 


1 .2989 


I«3l2I 


1 -3252 


60 



15 





TABLE 


OF CHORDS: 


[Radius 


=1.0000] 




i 


o» 


§3° 


84° 


85° 
i.35i2 


86° 


87° 


88° 


89° 


0' 


1.3252 


1.3383 


i-364o 


1.3767 


i-38 9 3 


1.4018 


I 


1-3255 


1.3385 


i.35i4 


1-3642 


1.3769 


i-38 9 5 


1 .4020 


1 


2 


1.3257 


1-3387 


i.35i6 


1.3644 


1. 3771 


i-38 9 7 


I-4022 


2 


3 


i 1.3259 


1.3389 


i.35i8 


1.3646 


i.3 77 3 


i.38 99 


I -4024 


3 


4 


1. 3261 


1-3391 


1.3520 


1.3648 


1.3776 


1 .3902 


I .4026 


4 


5 


1.3263 


i-33 9 3 


1-3523 


i.365i 


1.3778 


1-3904 


I .4029 


5 


6 


1-3265 


1.3396 


1-3525 


1.3653 


1.3780 


1 .3906 


i -4o3i 


6 


7 


1.3268 


1.3398 


1.3527 


i.C-655 


1.3782 


1.3908 


i-4o33 


7 


8 


1.3270 


i.34oo 


1 .3529 


i.365 7 


i.3 7 84 


1 .3910 


i-4o35 


8 


9 


1.3272 


1.3402 


i.353i 


1-3659 


1.3786 


1. 3912 


i-4o37 


9 


IO 

ii 


1.3274 
1.3276 


i-34o4 


1-3533 


i.366i 


1.3788 


1.3914 


1 .4039 


10 
11 


1.3406 


1-3535 


1.3663 


1.3790 


1 .3916 


i-4o4i 


12 


1.3279 


1 .3409 


1-3538 


1.3665 


1.3792 


1. 3 9 i8 


i.4o43 


12 


i3 


1.3281 


1. 34 ii 


1-3540 


1-3668 


i. 3 79 4 


1.3920 


i.4o45 


i3 


i4 


1-3283 


i-34i3 


1.3542 


1 .3670 


1.3797 


1 .3922 


1.4047 


i4 


i5 


1.3285 


i.34i5 


1.3544 


i.36 7 2 


1.3799 


1.3925 


1.4049 


i5 


16 


1.3287 


1. 3417 


1.3546 


i-36 7 4 


i-38oi 


1.3927 


i-4o5i 


if 


17 


1.3289 


1. 3419 


1.3548 


i.36 7 6 


i.38o3 


1.3929 


i-4o53 


17 


18 


1 .3292 


1. 3421 


1 «355o 


i.36 7 8 


i-38o5 


i.3 9 3i 


i-4o55 


18 


x 9 


1.3294 


1.3424 


1 .3552 


i.368o 


1.3807 


i-3 9 33 


i-4o58 


l 9 


20 
21 


1 .3296 


1-3426 


1.3555 


1.3682 
1-3685 


1.3809 


i-3 9 35 
i.3 9 3 7 


1 -4o6o 


20 
21 


1.3298 


1.3428 


1.3557 


i.38ii 


1.4062 


22 


1 .33oo 


i.343o 


1.3559 


1.3687 


i.38i3 


i.3 9 3 9 


1.4064 


22 


23 


1 -33o2 


1.3432 


i.356i 


1.3689 


i.38i6 


i-3 9 4i 


1.4066 


23 


24 


i«33o5 


1.3434 


1.3563 


1 .3691 


i-38i8 


1.3943 


1.4068 


24 


25 


1.3307 


1.3437 


1-3565 


1.3693 


1.3820 


i.3 9 45 


1-4070 


25 


26 


1.3309 


1.3439 


i-356 7 


1 -3695 


1.3822 


1 • 3947 


1.4072 


26 


27 


i.33n 


1.3441 


1.3570 


1.3697 


1.3824 


1 .3950 


1.4074 


27 


28 


i.33i3 


1-3443 


1 -3572 


1 -3699 


1.3826 


1 .3952 


1.4076 


28 


29 


i-33i5 


1-3445 


i-35 7 4 


1 -3702 


1.3828 


1-3954 


1.4078 


29 


36 

3i 


i-33i8 


1-3447 


1.3576 


1.3704 


i-383o 


1 -3936 


1.4080 


3o 
3i 


i.332o 


1.3449 


i-35 7 8 


i.3 7 o6 


1-3832 


i-3 9 58 


1.4082 


32 


1.3322 


1 .3452 


i.358o 


i.3 7 o8 


1-3834 


1 .3960 


1.4084 


32 


33 


1.3324 


1.3454 


1.3582 


1. 3710 


i-383 7 


1 .3962 


1.4086 


33 


M 


1.3326 


1-3456 


1.3585 


1 .371 2 


1 -3639 


i-3 9 64 


1 .4089 


34 


35 


1.3328 


1-3458 


1.3587 


i.3 7 i4 


1. 384i 


1.3966 


1.4091 


35 


36 


1. 333 1 


1 • 346o 


1 .3589 


i.3 7 i6 


1-3843 


i-3q68 


1 • 4093 


36 


3- 


1.3333 


1.3462 


1 -3591 


1. 3718 


1-3845 


1.3970 


1.4095 


3 7 


38 


1.3335 


i.3 4 65 


1 -3593 


1 .3721 


1-3847 


1.3972 


1.4097 


3S 


3 9 


1.3337 


1.3467 


i.35 9 5 


1.3723 


1-3849 


1 -3975 


1.4099 


3 9 


4o 
4i 


1.3339 


1.3469 


1.3597 


1.3725 
1.3727 


i-3t>5i 


1.3977 


i«4ioi 


4o 

4i 


1. 334i 


1. 3471 


1.3599 


1-3853 


1.3979 


i-4io3 


42 


1.3344 


1.3473 


1 -36o2 


1.3729 


1 -3855 


1. 3981 


i-4io5 


42 


43 


1.3346 


1.3475 


i-36o4 


i.3 7 3i 


1-3858 


1.8953 


1.4107 


43 


44 


1-3348 


1-3477 


1 >36o6 


i-3 7 33 


i-386o 


i .3985 


1. 4109 


44 


45 


i.335o 


i.348o 


i-36o8 


i-3 7 35 


I-3b02 


10967 


1 -4i 1 1 


45 


46 


1-3352 


1-3482 


1 -36io 


i-3 7 38 


I-3S64 


1 .3989 


i.4n3 


46 


47 


1-3354 


1.3484 


i.36i2 


i-3 7 4o 


1-3866 


1.3991 


i.4n5 


47 


48 


1-3357 


1.3486 


i-36r4 


1.3742 


1-3868 


1.3993 


1.4117 


48 


49 


1.3359 


1.3488 


i-36i 7 


i-3 7 44 


1.3870 


1-3990 


1.4119 


4 9 


5o 
5i 


i-336i 


1.3490 


1 .3619 


i-3 7 46 


1.3872 


1.3997 


1 .4122 


5o 
5i 


1.3363 


1 • 3492 


1. 362i 


i.3 7 48 


i.38 7 4 


1.3999 


1. 4124 


52 


1-3365 


1.3493 


1 -3623 


i.3 7 5o 


i.38 7 6 


1 .4002 


1. 4126 


52 


53 


1-3367 


1.3497 


1 -3625 


1.3752 


1.3879 


• 1 . 4oo4 


1. 4128 


53 


54 


1.3370 


1.3499 


t-3627 


i-3 7 54 


i.388i 


1 . 4006 


i-4r3o 


54 


55 


1.3372 


i.35oi 


1-3629 


1.3757 


1-3883 


1.4008 


i-4i32 


55 


56 


i.33 7 4 


i«35o3 


i-363i 


i-3 7 59 


1-3885 


1 .4010 


i-4i34 


56 


5 7 


1.3376 


i-35o5 


1-3634 


i-3 7 6i 


i-3S8 7 


1. 4012 


i.-ji36 


57 


58 


1.3378 


i-35o8 


1-3636 


1-3-63 


i-3S8 9 


I - -4O 1 4 


i-4i38 


55 


5 9 


1-3380 


1 «35io 


1-3638 


1.3765 


1 -3891 


I -4oi6 


1. 41-40 


^9 


60 


1-3383 


1-3512 


1 • 364o 


1.3-67 


1.3S93 


i«4ciS 


1. -41-42 


60 D 



16 



TABLE I.. 



LOGARITHMS OF NUMBERS 



1 to lOOOO. 



N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 


I 


o • oooooo 


26 


I -4*4973 
i-43i364 


5i 


l -707570 


76 


1 -880814 


2 


o-3oio3o 


3 


52 


1 .716003 


77 


1-886491 


3 


o-477'2i 


i-447i58 


53 


I.724276 


78 


1 -892095 


4 


0-602060 


29 


i-4623 9 8 


54 


I -732394 


79 


1-897627 


5 


0-698970 


3o 


1.477*21 


55 


i • 74o363 


80 


1 -903090 


6 


0-778151 


3i 


i-49i362 


56 


1. 748188 


81 


1-908485 


I 


0-845098 


32 


i-5o5i5o 


57 


1.755875 


82 


1 -913814 


0-903090 


33 


i-5i85i4 


58 


1.763428 


83 


1-919078 


9 


0-954243 


34 


1-53 1479 


5 9 


i'77o852 


84 


1-924279 


10 


1 -oooooo 


35 


1 • 544o68 


60 


1 -778151 


85 


1-929419 


ii 


1 -o4i393 


36 


i-5563o3 


61 


i-78533o 


So 


1 -934498 


12 


1 -079181 
1 -i i3943 


Si 


1-568202 


62 


1 -792392 


87 


1 .939519 


i3 


1-579784 


63 


1 -799341 


88 


1-944483 


14 


1-146128 


3 9 


1 -591065 


64 


r -806180 


89 


1-949390 


i5 


1-176091 


40 


1 -602060 


65 


1 -812913 


90 


I-954243 


16 


1 -204120 


4i 


1-612784 


66 


1-819544 


9i 


1 -959041 


17 


1 -23o449 

I -2552-73 


42 


1-623249 


67 


1 -826075 


92 


1-963788 


18 


43 


1-633468 


68 


i-8325o9 


9 3 


1-968483 


!9 


1-278754 


44 


1 -643453 


69 


1-838849 


94 


1 -973128 


20 


i-3oio3o 


45 


I-6532I3 


70 


1-845098 


9 5 


1-977724 


21 


1 -322219 
1-342423 


46 


1-662758 


71 


i-85i258 


96 


1-982271 


22 


47 


1-672098 


72 


1-857333 


11 


1-986772 


23 


1*361728 


48 


1-681241 


73 


1.863323 


1-991226 


24 


i-'38o2ii 


49 


1-690196 


74 


1-869232 
1-875061 


99 


1 -995635 


25 


1-397940 


5o 


1-698970 


75 


100 


2 • oooooo 



N. B. In the following table, in the last nine columns of each page, where thn 
Hrst or leading figures change from 9's to O's, the character ♦ is introduced instead 
of the O's, to catch the eye, and to indicate that from thence the annexed firs! 
two figures of the Logarithm in the second oolumn stand in the next, lower lino 
directly under the asterisk. 



37 



2 




LOGARITHMS OF NUMBERS. Table L 


ST. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


100 1 


00 0000 


o434 


0868 


»3oi 


1734 


2166 


25 9 8 


3029 


3461 


38 9 i 


43a 


101 


4321 


475i 


5i8i 


t>6o9 


6o38 


6466 


6894 


7321 


7748 


8174 


428 


102 


*86oo 


9026 
3259 


Q45 1 
368o 


9876 


♦3 00 


0724 


1 147 


1570 


1993 


24i5 


424 


io3 


01 283 7 


4100 


4521 


4940 


536o 


5779 


6197 


6616 


419 


io4 


*7o33 


745 1 


7868 


8284 


8700 


9116 


9532 


9947 


♦36i 


0775 


416 


io5 


02 1 1 89 


i6o3 


20:6 


2428 


2841 


3252 


3664 


4075 


4486 


4896 


412 


ID6 


53o6 


5 7 i5 


6i25 


6533 


6942 


7350 


7757 
1812 


8164 


8571 


8978 


408 


107 


*9384 


Vn 89 

3826 


♦195 


0600 


1004 


1408 


2216 


2619 


3021 


404 


ic8 


o3 3424 


4227 


4628 


5029 


543o 


583o 


623o 


6629 


7028 


400 


109 


* 7426 


7825 


8223 


8620 


9017 


94i4 


981 1 


♦207 


0602 


0998 


3 9 6 


no 


04 1393 


1787 


2182 


25 7 6 


2969 

6885 


3362 


3 7 55 
7664 


4i48 


454c 


4932 

883o 


3 9 3 

38 9 


III 


5323 


5714 


6io5 


64o5 


7273 


8o53 


8442 


112 


♦ 9218 
o5 3078 


9606 
3463 


ag 


♦38o 


0766 


n53 


1 538 


1924 


2309 


2694 


386 


n3 


423o 


46i3 


4996 


53 7 8 


5760 


6142 


6524 


382 


114 


#6905 


7286 


7666 


8046 


8426 


88o5 


9 i85 


9 563 


9942 


♦320 


379 


n5 


06 0698 
4458 


1075 


1452 


1829 


2206 


2582 


2958 


3333 


3709 


4o83 


3 7 6 


116 


4832 


52o6 


558o 


5 9 53 


6326 


6699 


7071 


7443 


78i5 


372 


117 


*8i86 


8557 


8928 


9298 


9668 
3352 


♦o38 


0407 


0776 


n45 


i5i4 


36 9 


118 


07 1882 


2250 


2617 


2985 


3 7 i8 


4o85 


445 1 


4816 


5i82 


366 


119 


5547 


5912 


6276 


6640 


7004 


7368 


773i 


8094 


8457 


8819 


363 


120 


* 9181 


9543 

3 144 


9904 
3oo3 


♦266 


0626 


0987 


1 347 


1707 


2067 


2426 


36o 


121 


08 2785 


386 1 


4219 


4376 


4934 


5291 


5647 


6004 


35 7 


122 


636o 


6716 


7071 


7426 


7781 


8i36 


8490 


8845 


9198 


o552 
J071 


355 


123 


*99o5 


♦258 


061 1 


0963 


i3i5 


1667 


2018 


2370 


2721 


35i 


124 


093422 


3772 


4122 


4471 


4820 


5169 


55i8 


5866 


62i5 


6562 


349 


125 


#6910 


7 25 7 


7604 


7 9 5i 


8298 


8644 


8990 


9 335 


0681 
J119 


♦026 


346 


126 


10 0J71 


0715 


1059 


Uo3 


1747 


2091 


2434 


2777 


3462 


343 


127 


38o4 


4U6 


4487 
7888 


4828 


5169 


55io 


585 1 


6191 


653 1 


6871 


340 


128 


*72I0 


7549 


8227 


8565 


8 9 o3 


9241 


9579 


9916 
3275 


♦253 


338 


129 


II 0590 


0926 


1263 


1 5 99 


1934 


2270 


26o5 


2940 


3609 


335 


i3o 


3 9 43 


4277 


461 1 


4944 


52 7 8 


56n 


5943 


6276 


6608 


6940 


333 


i3i 


#7271 


7603 


79 34 


8265 


85 9 5 


8926 


Q256 


9 586 


99i5 


♦245 


33o 


132 


12 0574 


0903 


I23l 


i56o 


1888 


2216 


2544 


2871 


3i 9 8 


3525 


328 


133 


3852 


4178 


45o4 


483o 


5i56 


548i 


58o6 


6i3i 


6456 


6781 


325 


1 34 


*7io5 


7429 


7753 


8076 


8399 


8722 


9045 


9 368 


9690 


♦012 


323 


i35 


i3 o334 


o655 


0977 


1298 


1619 


1939 
5i33 


2260 


258o 


2900 


3219 

64o3 


321 


1 36 


353 9 


3858 


4177 


4496 


48i4 


545 1 


5769 


6086 


3i8 


i3 7 


6721 


7037 


7354 


7671 


7987 


83o3 


8618 


8934 


9249 


9564 


3i5 


1 38 


*o8 7 9 
i4 3oi5 


♦ 194 


o5o8 


0822 


n36 


U5o 


1763 


2076 


238 9 


2702 


3i4 


i3 9 


3327 


3639 


3951 


4263 


4574 


4885 


5196 


55o7 


58i8 


3n 


140 


6128 


6438 


6748 


7 o58 


7 36 7 


7676 


79 85 


8294 


86o3 


891 1 


3og 


141 


*92i9 

152288 


9 52 7 


9 835 


♦ 142 


0449 


0756 


io63 


1370 


1676 


1982 


307 


142 


25g4 


2900 


32o5 


35io 


38i5 


4120 


4424 


4728 


5o32 


3o5 


143 


5336 


5640 


5943 


6246 


6549 


6852 


7i54 


7457 


7759 


8061 


3o3 


144 


*8362 


8664 


8965 


9266 


9 56 7 


9868 


♦168 


0469 


0769 


1068 


3oi 


145 


16 i368 


1667 


1967 


2266 


2564 


2863 


3i6i 


3460 


3 7 58 


4o55 


299 


146 


4353 


465o 


4947 


5244 


554i 


5838 


6i34 


643o 


6726 


7022 


297 


148 


73i 7 


76i3 


7908 


82o3 


8497 
1434 


8792 


9086 


9 38o 


9674 


0968 
2895 


295 


17 0262 


o555 


0848 


1141 


1726 


2019 


23 11 


26o3 


293 


149 


3i86 


3478 


3769 


4060 


435 1 


4641 


4932 


5222 


55i2 


5So2 


291 


i5o 


6091 


638i 


6670 


6959 
9839 


7248 


7536 


7825 


8n3 


8401 


8689 
1 558 


289 


i5i 


*8o 7 7 
18 1844 


9264 


9552 


♦ 126 


o4i3 


0699 


o 9 85 


1272 


-87 


i5a 


2129 


24i5 


2700 


2o85 


3270 


35dd 


3S3 9 


4123 


4407 


285 


i53 


4691 


4973 
78o3 


5259 


5542 


5825 


6108 


6391 


6674 


69D6 


7239 


283 


1 54 


* 7521 


8084 


8366 


8647 


8928 


9209 


9490 


9771 


♦o5i 


2S1 


155 


19 o332 


0612 


0892 
368i 


1171 


I45i 


i73o 


2010 


2289 


2567 
5346 


2846 


9 


1 56 


3i25 


34o3 


3g5g 


423 7 


45i4 


4792 


5069 


5623 


1 57 


5900 


6176 


6453 


6729 


7oo5 


7281 


75d6 


7832 


S107 


8382 


276 


1 58 


*8657 


8 9 3 2 


9206 


948i 


9755 


♦029 


o3o3 


0577 


o85o 


1 1 24 


274 


i5g 


20 1 397 


1670 


1943 


2216 


2488 


2761 


3o33 


33o5 


35 77 


3848 


272 


u 





1 | 2 


8 


4 


5 


6 


7 


8 | 9 


D. 



Table I. 


LOGARITHMS OF NUMBERS. 3 


M". 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


1 60 


204120 


4391 


4663 


4934 


5204 


5475 


5746 


6016 


6286 


6556 


271 


161 


6826 


7096 
97 83 


7365 


7634 


7904 
o586 


8i 7 3 


8441 


8710 


8979 


9247 


269 


162 


♦ 95 1 5 


♦o5i 


o3i9 


o853 


1121 


1 388 


1 654 


1921 
4579 


267 
266 


i63 


21 2188 


2454 


2720 


2986 


3252 


35i8 


3783 


4049 


43i4 


164 


4844 


5109 


53 7 3 


5638 


5902 


6166 


643o 


6694 


6 9 5 7 


7221 


264 


165 


7484 


7747 


8010 


8273 


8536 


8798 


9060 


g323 


9 585 


9846 


262 


166 


22 0108 


0370 


o63[ 


0892 


n53 


I4U 


1675 


1936 
4533 


2196 


2456 


261 


l6 7, 
168 


2716 


2976 
5568 


3236 


3496 
6084 


3755 


4oi5 


4274 


4792 


5o5i 


259 

258 


5309 


5826 


5342 


6600 


6858 


71 15 


7372 


763o 


169 


#7887 


8i44 


8400 


8657 


8 9 i3 


9170 


9426 


9682 


9938 


♦ig3 


256 


170 


i3 0449 


0704 


0960 
35o4 


I2l5 


1470 


1724 


1979 
4517 


2234 


2488 


2742 


254 


171 


2996 
55 2 8 


325o 


3 7 5 7 


401 1 


4264 


4770 


5o23 


5276 


253 


172 


5 7 8i 


6o33 


6285 


6537 


6789 


7041 


7292 


7544 


7 79 5 


25a 


173 


*8o46 


8297 


8548 


8799 


9049 


9299 
I7 9 5 


955o 


9800 


♦o5o 


o3oo 


25o 


174 


24 0649 


0799 


1048 


1297 


1 546 


2044 


2293 


254i 


2790 


249 


175 


3o38 


3286 


3534 


3782 


4o3o 


4277 

6745 


4525 


4772 


5019 


5266 


248 


176 


55i3 


5759 


6006 


6252 


6499 
8o54 
IJ95 


6991 


7237 


7482 


7728 


246 


3 


*7973 


8219 


8464 


8709 


9198 
i638 


9443 


9687 


99 32 

2368 


♦ 176 


245 


25 0420 


0664 


0908 
3338 


ii5i 


1881 


2125 


2610 


243 


179 


2853 


3096 


358o 


3822 


4064 


43o6 


4548 


4790 


5o3i 


242 


180 


52 7 3 


55i4 


5 7 55 


5gg6 
83 9 8 
0787 


6237 


6477 


6718 


6958 

9 355 


7198 


743o 
9 833 


241 


181 


7679 


7018 
o3io 


81 58 


8637 

1025 


8877 


91 16 


9594 


23g 
238 


182 


26 007 1 


o548 


1263 


i5oi 


1739 


1976 


2214 


1 83 


245 1 


2688 


2923 


3i62 


3399 


3636 


38 7 3 


4109 


4346 


4582 


23" 


184 


4818 


5o54 


5290 


5525 


576 1 


5996 


6232 


6467 


6702 


6 9 3 7 


235 


181 


7172 


74o6 


7641 


7875 


Suo 


8344 


8578 


8812 


9046 


9279 


234 


186 


*95i3 


9746 


9980 


♦ 2l3 


0446 


0679 


0912 


1 144 


1377 


1609 


233 


187 


27 1842 


2074 


23o6 


2538 


2770 


3ooi 


3233 


3464 


36 9 6 


3927 


232 


188 


4i58 


4389 


4620 


485o 


5o8i 


53u 


5542 


5772 


6002 


6232 


23o 


189 


6462 


6692 


6921 


7i5i 


738o 


7609 


,7838 


8067 


8296 


8025 


229 


190 


#8754 


8982 


9211 


943o 
1710 


9667 


9895 


♦ 123 


o35i 


0578 


0806 


228 


iq* 


28 io33 


1261 


1488 


1942 


2169 


2396 
4656 


2622 


2849 


3075 


227 


192 


33oi 


3527 


3 7 53 


3979 


42o5 


443 1 


4882 


5107 


5332 


226 


193 


555 7 


5782 


6007 


6232 


6456 


6681 


6905 


7i3o 


7354 


i5 7 8 


225 


194 


7802 


8026 


8249 


8473 


8696 


8920 


9i43 


9366 


9589 


9812 


223 


195 


29 oo35 


0257 


0480 


0702 


0925 


1 147 
3363 


1369 


1691 


i8i3 


2o34 


222 


196 


2256 


2478 


2699 


2920 


3i4i 


3584 


38o4 


4025 


4246 


221 


197 


4466 


4687 


4907 


5127 


5347 


5567 


5787 


6007 


6226 


6446 


220 


198 


6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8635 


219 
2l8 


199 


*8853 


9071 


9289 


9507 


9725 


9943 


♦ 161 


0378 


o5g5 


o8i3 


50O 


3o io3o 


1247 


1464 


1681 


1898 
4069 


2114 


233i 


2547 


2764 


2980 


217 


201 


3196 


34i2 


3628 


3844 


4275 


4491 
663o 


4706 
6854 


4921 


5i36 


2l6 


202 


53oi 


5566 


5 7 8i 


5996 


6211 


6425 


7068 


7282 


2l5 


203 


7496 
*963o 


7710 


7924 


8137 
0268 


835i 


8564 


8778 


8991 


9204 


9417 


213 


204 


9843 


♦o56 


0481 


0693 


0906 


1118 


i33o 


1 542 


212 


205 


3i 1754 


1966 


2177 
4289 


238 9 


2600 


2812 


3o23 


3234 


3445 


3656 


211 


206 


3867 


4078 


4499 


4710 


4920 


5i3o 


5340 


555i 


5760 


210 


207 


5970 


6180 


63go 

8481 


65 99 


6809 
8898 


7018 


7227 


7436 


7646 


7854 


209 
208 


208 


8o63 


8272 


8689 


9106 


g3i4 


9522 


9730 
i8o5 


9938 


209 


32 0146 


o354 


o562 


0769 


0977 


1 184 


1391 


1 5 9 8 


2012 


207 


210 


2219 


2426 


2633 


283 9 


3o46 


3252 


3458 


3665 


3871 


4077 


206 


211 


4282 


4488 


4694 


4899 
6930 


5io5 


53io 


55i6 


5721 


5926 


6i3i 


2o5 


212 


6336 


654i 


6745 


7i55 


735o 
9398 


7563 


7767 


7972 


8176 


204 


213 


*838o 


8583 


8787 
0819 


8991 


9194 


9601 


9 8o5 


♦008 


0211 


203 


214 


33 0414 


0617 


1022 


1225 


1427 


i63o 


i832 


2o34 


2236 


202 


2l5 


2438 


2640 


2842 


3o44 


3246 


3447 
5458 


3649 
5658 


385o 


4o5i 


4253 


202 


216 


4454 


4655 


4856 


5o57 


5257 


585 9 
7858 


6059 
8o58 


6260 


201 


£2 


6460 


6660 


6860 


7060 


7260 


7459 


765 9 


8257 
0246 


200 


*8456 


8656 


8855 


9054 


9253 


945 1 


965o 


9849 


♦047 
2028 


199 
I98 


219 


34 0444 


0642 


0841 


1039 


1237 


1435 


i632 


i83o 


2225 


N. 





1 


2 


3 


4 


1 5 


6 


7 


8 


9 


D. 



4 




LOGARITHMS OF NUMBERS. Table L 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


220 


34 2423 


2620 


2817 
4785 


3oi4 


3212 


3409 


36o6 


38o2 


3999 


4196 


197 


221 


43q2 


4589 


4981 


5i 7 « 


"74 


5370 


5766 


5962 


6137 


196 


222 


6353 


6549 


6744 


6939 

8889 


7i35 


733o 


7525 


7720 


7915 
9860 


8110 


193 


223 


*83o5 


85oo 


8694 
o636 


9083 


9278 


9472 


9666 


♦o54 


194 


224 


35 0248 


0442 


0829 


1023 


1216 


1410 


i6o3 


1796 


1989 


i 9 3 


325 


2i83 


23 7 5 


2568 


2761 


2954 

4876 


3i47 


333 9 


3532 


3724 


3916 
5834 


i 9 3 


226 


4108 


43oi 


4493 


4685 


5o68 


526o 


5452 


5643 


192 


227 
228 


6026 


6217 


6408 


6599 
85o6 


6790 


6981 
8886 


7172 


7363 


7354 


7744 


191 


79 35 


8i25 


83i6 


8696 


9076 


9266 


9456 


9646 


a; 


229 


* 9 835 


♦025 


02l5 


0404 


0593 


0783 


0972 


1161 


i35c 


i539 


23o 


361728 


IOI7 


2105 


2294 


2482 


2671 


285g 


3o48 


3236 


3424 


188 


23l 


36i2 


38oO 


3 9 S8 


4176 


4363 


455 1 


4739 


4926 


5n3 


53oi 


1 83 


232 


5488 


5675 


5862 


6049 


6236 


6423 


6610 


6796 


6 9 83 
8845 


7169 


187 


233 


7356 


7542 


7729 
9 58 7 


7913 


8101 


8287 


8473 


8639 

03l3 


9o3o 


186 


234 


«• 9216 


9401 


9772 


9958 


♦143 


o328 


0698 


o883 


i85 


235 


37 1068 


1253 


1437 


1622 


1806 


1991 


2175 


236o 


2544 


2728 


184 


236 


2912 


3096 
4932 


3280 


3464 


3647 


383 1 


4oi5 


4198 


4382 


4565 


184 


23 7 


4748 


5n5 


5298 


548i 


5664 


5846 


6029 


6212 


63 9 4 


183 


238 


6577 


6 7 5 9 


6942 


7124 


73o6 


7488 


7670 


7852 


8o34 


8216 


182 


239 


*83 9 8 


858o 


8761 


8943 


9124 


93o6 


9487 


9668 


9849 


♦o3o 


181 


240 


380211 


0392 


o5 7 3 


0754 


0934 


iii5 


1296 


1476 


1 656 


i83 7 


181 


241 


2017 


2197 


2377 


2537 


2 7 3 7 


2917 


3o 97 


3277 


3456 


3636 


180 


242 


38i5 


3995 

5 7 85 


4i74 


4353 


4533 


4712 


4891 


5070 


5249 


5428 


; 1 


243 


56o6 


5964 


6142 


6321 


6499 


6677 


6856 


7034 


7212 
8989 


244 


7390 


7568 


7746 


7923 


8101 


8279 


8456 


8634 


881 1 


„8, 


245 


*9i66 


9 343 


95 oo 


9698 


9 8 7 5 


♦o5i 


0228 


o4o5 


o582 


0759 


'% 


240 


39 0935 


1112 


1288 


1464 


1641 


1817 


1993 


2169 


2345 


2521 


247 


2697 


2873 


3o48 


3224 


34oo 


35 7 5 


3 7 3I 


3926 


4101 


4277 
6025 


176 


248 


4432 


4627 


4802 


4977 


5i52 


5326 


55oi 


56 7 6 


585o 


I 7 D 


249 


6199 


63 7 4 


6548 


6.722 


6896 


7071 


7245 


7419 


7592 


7766 


174 


25o 


7940 


8114 


8287 


8461 


8634 


8808 


8981 


9i54 


9328 


9501 


I 7 3 


25l 


#9674 


9847 


♦020 


0192 


o365 


o538 


071 1 


o883 


io56 


1228 


I 7 3 


252 


4o 1401 


i573 


1745 


1917 


2089 


2261 


2433 


26o5 


2777 


2 9 40 

4663 


172 


253 


3 1 2 1 


3292 


3464 


3635 


3807 


3 97 8 
5688 


4149 

5858 


4320 


4492 


HI 


254 


4834 


5oo5 


5i 7 6 


5346 


55i7 


6029 


6199 


6370 


171 


255 


6540 


6710 


6881 


7o5i 


7221 


73oi 
9087 


756i 


773i 


7901 


8070 


170 


256 


8240 


8410 


8579 


8749 


8918 
0600 
2 29 6 


9257 


9426 


9393 
1283 


9764 


I69 


257 


*9933 


♦102 


0271 


0440 


0777 


0946 


1114 


i45i 


3 


258 


41 1620 


1788 


1956 


2124 


2461 


2629 


2796 


2964 


3i32 


259 


33oo 


3467 


3635 


38c3 


3970 


4i37 


43o3 


4472 


463g 


4806 


167 


260 


4973 


5i4o 


53o7 


5474 


564i 


58o8 


5 97 4 
7 638 


6141 


63o8 


6474 


167 


261 


6641 


6807 


6973 


7i3o 

8798 


73o6 


7472 


7804 


7970 


8i35 


166 


262 


83oi 


8467 


8633 


8964 


9129 


9293 


9460 


9625 


9791 
i43 9 


i65 


263 


*9956 


♦121 


0286 


0431 


0616 


0781 


0943 


mo 


1275 


i65 


264 


42 1604 


1768 


1933 


2097 


2261 


2426 


2390 


2754 


2918 


3o82 


164 


265 


3246 


34io 


35 7 4 


3737 


3901 


4o65 


4228 


4392 


4555 


4718 


164 


266 


4882 


5o45 


52o8 


5371 


5334 


5697 


586o 


6o23 


6186 


6349 
7973 


1 63 


267 
268 


65u 


6674 


6836 


6999 


7161 

8 7 83 


7324 


7486 


7648 


7811 


162 


8i35 


8297 


8459 


8621 


8944 


9106 


9 ^ 8 


9429 


9391 


162 


269 


*9752 


9914 


♦070 


0236 


0398 


0339 


0720 


0881 


1042 


1203 


161 


270 


43 i364 


i525 


1685 


1846 


2007 


2167 


2328 


2488 


2649 


2809 


161 


271 


2969 


3i3o 


3290 
4888 


345o 


36io 


3770 


3930 


4090 
5685 


4249 


4409 


160 


272 


4269 
6i63 


4729 


5o48 


5207 


5367 


5326 


5844 


6004 


1 5 9 


2 7 3 


6322 


6481 


6640 


tsn 


6 9 5 7 


7116 


7275 
885 9 


7433 


7592 


d 


374 


7 7 5i 


7909 


8067 


8226 


8342 


8701 


9017 


9175 


275 


* 9333 


9491 


9648 


e<*o6 


0964 


♦ 122 


0279 


0437 


0594 
2166 


0752 


1 58 


276 


440909 


1066 


1224 


i38i 


1338 


1695 


1832 


2009 
3576 


2323 


1 5 7 


3 


2480 


2637 


2793 


2950 


3io6 


3263 


3419 


3 7 32 


3S89 


i5 7 


4o45 


4201 


4337 


43i3 


4669 


4825 


4981 


5i37 


5293 


5440 
700J 


1 56 


279 


56o4 


5760 


5gi5 6071 


6226 


6382 


6337 


6692 


6S48 


i55 
D. 


N. 





1 


2 | 3 | 4 


5 


6 


7 


8 


9 



Table I. 


LOGARITHMS OF NUMBERS. 5 


N. 





1 


2 I 


3 


4 1 


5 


6 


7 


8 


9 D. 


280 


44 7i58 


7 3i3 

8861 


7468 


7623 


7778 


7 9 33 


8088 


8242 


83 9 7 


8552 


1 55 


281 


#8706 


9015 


9170 


9324 


9478 


;633 


n*i 


9941 


♦095 


1 54 


282 


45 0249 


o4o3 


o557 
2093 


07 1 » 


o865 


1018 


1172 


i326 


1479 


i633 


1 54 


283 


1786 


1940 


2247 


2400 


2553 


2706 


285 9 


3012 


3i65 


1 53 


284 


33i8 


3471 


3624 


3777 


3g3o 


4082 


4235 


438 7 


4540 


4692 


153 


285 


4845 


tm 


5i5o 


53o2 


5454 


56o6 


5 7 58 


5910 


6062 


62:4 


l52 


286 


6366 


6670 


682! 


6973 


7125 

8638 


7276 
8789 


7428 


7579 


773i 


l52 


287 


7882 


8o33 


8184 


8336 


8487 


8940 


9091 


9242 


i5i 


288 


♦ 9392 


9543 


9694 


9845 


9995 


♦ 146 


0296 


0447 


o5q7 


0748 


i5i 


289 


46 0898 


1048 


1198 


i348 


1499 


1649 


1799 


1948 


2098 


2248 


i5o 


290 


2398 


2548 


2697 


2847 


2997 


3i46 


3296 


3445 


35 9 4 


3744 


i5o 


291 


38 9 3 
5383 


4042 


419 1 


434o 


4490 


4639 


4788 


4936 


5o85 


5234 


149 


292 


5532 


568o 


5829 


5977 


6126 


6274 


6423 


65 7 i 


6719 


149 
148 


293 


6868 


7016 
8495 


7164 


73i2 


7460 


7608 


7756 


7904 
9 38o 


8o52 


8200 


294 


8347 


8643 


8790 


8 9 38 


9085 


9233 


9527 


9675 


148 


295 


4*9822 


t& 


♦ 116 


0263 


0410 


o557 


0704 


o85i 


0998 


IU5 


147 


296 


471292 

2756 


i585 


1732 


1878 


2025 


2171 


23i8 


2464 


2610 


146 


297 


2903 
4362 


3 049 
45o8 


3195 
4653 


334i 


3487 


3633 


3779 


3925 


4071 


146 


298 


4216 


4799 

6252 


4944 


5090 


5235 


538i 


5526 


146 


299 


5671 


58i6 


5962 


6107 


63 97 


6542 


6687 


6832 


6976 


145 


3oo 


7121 


7266 


741 1 


7555 


7700 


7844 


7989 


8i33 


8278 


8422 


145 


3oi 


8566 


8711 


8855 


Ss 


9143 


9287 
0725 


943 1 


9 5 7 5 


97'9 


9 863 


144 


302 


48 0007 


oi5i 


0294 


o582 


0869 


1012 


n56 


1299 

2731 


144 


3o3 


1443 


1 586 


1729 


1872 


2016 


2159 


2302 


2445 


2588 


143 


3o4 


2874 


3oi6 


3i5g 


33o2 


3445 


358 7 


373o 


38 7 2 


401 5 


4i 57 


143 


3o5 


43oo 


4442 


4585 


4727 


4869 


5ou 


5i53 


5295 


5437 


5579 


142 


3o6 


5721 


5863 


6oo5 


6147 


6289 


643o 


6572 


6714 


6855 


6997 


142 


3o7 
3o8 


7 i38 
855i 


7280 


7421 
8833 


7563 


7704 


7845 


7086 


8127 


8269 


8410 


141 


8692 


8974 
o38o 


9114 


9255 


9 3 9 6 


9537 


9677 


9818 


i4i 


309 


#9958 


♦099 


0239 


0D20 


0661 


0801 


0941 


1081 


1222 


140 


3io 


49 1 362 


l502 


1642 


1782 


I022 
33l9 


2062 


2201 


234i 


2481 


2621 


140 


3n 


2760 


2900 


3o4o 


3179 


3458 


3597 
4989 


3737 


38 7 6 


401 5 


139 


312 


4i55 


4294 


4433 


4572 


471 1 


485o 


5i28 


5267 


54o6 


139 


3i3 


5544 


5683 


5822 


5o6o 


58 


6238 


6376 


65i5 


6653 


6791 


i3q 
i38 


3i4 


6930 


7068 


7206 


7344 


7621 


77 5 9 


7897 


8o35 


8173 


3i5 


83u 


8448 


8586 


8724 


8862 


8999 


9 i3 7 


9275 


9412 


955o 


i38 


3i6 


4*9687 


9824 


9962 
i333 


♦099 


0236 


0374 


o5n 


0648 


0785 


0922 


i3 7 


3i 7 


5o 1059 


1 196 


1470 


1607 
2o 7 3 
4335 


1744 


1880 


2017 


2i54 


2291 
3655 


i3 7 


3i8 


2427 


2564 


2700 


283 7 


3109 


3246 


3382 


35i8 


1 36 


319 


3791 


3927 


4o63 


4199 


4471 


4607 


4743 


4878 


5oi4 


1 36 


320 


5i5o 


5286 


5421 


555 7 


56 9 3 


5828 


5 9 64 


6099 


6234 


6370 


i36 


321 


65o5 


6640 


6776 


691 1 


7046 
83 9 5 


7181 


7 3i6 


745 1 


7586 


7721 


i35 


322 


7856 


7991 


8126 


8260 


853o 


8664 


8799 
oi43 


8 9 34 


9068 


i35 


323 


*92o3 


9 33 7 


947i 


9606 


9740 


9874 


♦009 


0277 


0411 


1 34 


324 


5i o545 


0679 


o8i3 


0947 


1081 


I2l5 


i349 


1482 


1616 


1750 


1 34 


325 


1 883 


2017 


2l5l 


2284 


2418 


255i 


2684 


2818 


2951 


3o84 


i33 


326 


32i8 


335i 


3484 


36i 7 


375o 


3883 


4016 


4U9 


4282 


44i4 


1 33 


327 
328 


4548 


4681 


48i3 


4946 


5o 7 o 
64o3 


521 1 


5344 


5476 


5609 


5741 


1 33 


58 7 4 


6006 


6139 


6271 


6535 


6668 


6800 


6932 


7064 


132 


329 


7196 


7328 


7460 


7 5 9 2 


7724 


7 855 


7987 


8119 


825i 


8382 


132 


33o 


85 1 4 


8646 


8777 


8909 


qo4o 


9171 

0484 


93 o3 


9434 


9 566 


9697 


i3i 


33 1 


♦ 9828 


99 5 9 


♦090 


0221 


o353 


o6i5 


0745 


0876 


1007 


i3i 


332 


52 n38 


1269 
2575 


1400 


i53o 


1661 


1792 


1922 


2o53 


2i83 


23i4 


i3i 


333 


2444 


2705 


2835 


2966 


3096 


3226 


3356 


3486 


36i6 


i3o 


334 


3746 


3876 


4006 


4i36 


4266 


4396 


4526 


4656 


4785 


49i5 


i3o 


335 


5o45 


5i74 


53o4 


5434 


5563 


56 9 3 
6 9 85 


5822 


5951 


6081 


6210 


129 


336 


6339 


6469 


65 9 8 


6727 


6856 


71 14 

8402 


7243 


7 3 7 2 


75oi 


129 


33 7 
338 


763o 


7 7 5o 
9045 


7888 


8016 


8i45 


8274 


853 1 


8660 


8788 


129 

128 


.8917 


9 \li 
0456 


9302 


943o 


955q 


9687 


9 8i5 


9943 


♦072 


33 9 


53 0200 


o328 


o584 


0712 


0840 


0908 


1096 


1223 


i35i 


128 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



6 




LOGARITHMS OF NUMBERS. Table I. 


N. 
34o 



33 1479 


1 


2 


3 


4 


5 


6 1 7 


8 


9 


D. 


1607 


1734 


1862 


1990 


2117 


2245 1 2372 


25oo 


2627 


128 


34i 


2754 


2882 


3009 


3i36 


3264 


3391 


3oi8 


3645 


3772 


38 99 


127 


342 


4026 


41 53 


4280 


4407 


4534 


4661 


4787 


49U 


5o4i 


5167 


127 


343 


5294 


5421 


5547 


56 7 4 


58oo 


5 9 2 7 


6o53 


6180 


63o6 


6432 


126 


344 


6558 


6685 


681 1 


6 9 3 7 


7063 


7189 


73i5 


7441 


756 7 


7693 


126 


345 


7819 


7945 


8071 


8197 


8322 


8448 


85 7 4 


8699 


8825 


8951 


126 


346 


*9076 


9202 


9327 


9 452 


9378 


9703 


9829 


99 D4 


♦079 


0204 


125 


347 


540329 


0455 


o58o 


0705 


o83o 


0955 


1080 


1205 


i33o 


1454 


125 


348 


1579 


1704 


1829 


1953 


2078 


2203 


2327 


2452 


2576 


2701 


125 


349 


282D 


2900 


3074 


3i 99 


3323 


3447 


3571 


36 9 6 


3820 


3944 


124 


35o 


4o68 


4192 


43i6 


444o 


4564 


4688 


4812 


4936 


5o6o 


5i83 


124 


35i 


5307 


543 1 


5555 


56 7 8 


58o2 


5925 


6049 


6172 


6296 


6419 


124 


352 


6543 


6666 


6789 


6913 


7o36 


7l5 9 


7282 


74o5 
8635 


7529 


7652 


123 


353 


7775 


7898 


8021 


8i44 


8267 


838 9 


85i2 


8 7 58 


8881 


123 


354 


*90o3 


9126 


9249 


9 3 7 i 


9494 


9616 


9739 


9861 


9984 


♦106 


123 


355 


55o228 


o35i 


0473 


0595 


0717 
ig38 


0840 


0962 


1084 


1206 


i328 


122 


356 


i45o 


1572 


1694 


1816 


2060 


2181 


23o3 


2425 


2547 


122 


35 7 


2668 


2790 


2911 


3o33 


3i55 


3276 


33 9 8 


35i 9 


3640 


3762 


121 


358 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


473i 


4852 


4973 


121 


35q 


5094 


52i5 


5336 


5457 


5578 


5699 


5820 


5940 


6061 


6182 


121 


36o 


63o3 


6423 


6544 


6664 


6 7 85 


6905 


7026 


7146 


7267 


7 38 7 
858 9 


120 


36i 


75o7 


7627 
8829 


7748 


7868 


7988 


8108 


8228 


8349 


8469 


120 


362 


8709 


8948 


9068 


9188 


9 3o8 


9428 


9548 


9667 


9787 


120 


363 


*990 7 


♦026 


0146 


0265 


o385 


o5o4 


0624 


0743 


o863 


0982 


119 


364 


56 iioi 


1221 


1 340 


1459 


i5 7 8 


1698 


1817 


1936 


2o55 


2174 


119 


365 


2293 


2412 


253i 


265o 


2769 


2887 


3oo6 


3i25 


3244 


3362 


119 


366 


348i 


36oo 


3 7 i8 


383 7 


3903 


4074 


4192 


43u 


4429 


4548 


::? 


367 


4666 


4784 


4903 


5021 


5i3 9 


5257 


53 7 6 


5494 


56i2 


573o 


368 


5848 


5 9 66 


6084 


6202 


6320 


6437 


6555 


66 7 3 


6791 


6909 


ii8 


36g 


7026 


7144 


7262 


7379 


7497 


7614 


77 32 


7849 


7967 


8084 


118 


370 


8202 


8319 


8436 


8554 


8671 


8788 


8905 


9023 


9140 


9257 


"7 


3 7 i 


*9 3 74 


9491 


9608 


9725 


9842 


99 5 9 


♦076 


0193 


0309 


0426 


117 


I 12 


57 o543 


0660 


0776 


0893 


1010 


1 1 26 


1243 


i3do 

2523 


1476 


1592 


\\l 


373 


1709 

2872 


1825 


1942 


2008 


2174 


2291 


2407 


2639 


2735 


374 


2988 


3io4 


3220 


3336 


3452 


3568 


3684 


38oo 


3 9 i 5 


116 


3 7 5 


4o3i 


4U7 


4563 


4379 
5534 


4494 


4610 


4726 


4841 


4957 


5072 


116 


376 


5i88 


53o3 


5419 


56do 


5765 


588o 


5996 


6111 


6226 


n5 


111 


6341 


6457 


6572 


6687 


6802 


6917 


7o32 


7i47 
8295 


7262 


7377 
85 2 5 


n5 


7492 
863 9 


7607 


7722 


7 836 


795i 


8066 


8181 


8410 


u5 


3 79 


8754 


8868 


8 9 83 


9097 


9212 


9326 


9441 


9555 


9669 


114 


38o 


* 97 84 


9898 
1039 


♦012 


0126 


0241 


o355 


0469 
1608 


o583 


0607 
1 836 


081 1 


M4 


38i 


58 0925 


u53 


1267 


i38i 


1495 


1722 

2858 


1950 


114 


382 


2o63 


2177 


2291 


2404 


25i8 


263 1 


2745 


2972 


3o85 


n4 


383 


3i 99 
433 1 


33i2 


3426 


3539 


3652 


3765 


3879 


3992 


4io5 


4218 


1 13 


384 


4444 


4557 


4670 


4783 


4896 


5009 


5l22 


5235 


5348 


n3 


385 


546i 


5574 


5686 


5799 


5912 


6024 


6i3 7 


625o 


6362 


6475 


n3 


386 


6D87 


6700 
7823 
8q44 


6812 


6925 


7o37 


7U9 


7262 


7374 
8496 


7486 


75Q9 


112 


387 


771" 


7 9 35 


8047 


8160 


8272 


8384 


8608 


8720 


112 


388 


883? 


90D6 


9167 


9279 


9391 


95o3 


9615 


9726 


9 838 


112 


38 9 


» 9950 


♦061 


0173 


0284 


0396 


0507 


0619 


0730 


0842 


0953 


112 


390 


59 io65 


11-76 


1287 


1399 


i5io 


1621 


1732 


1843 


i 9 55 


2066 


111 


391 


2177 


2288 


2 3 99 
3oo8 


25lO 


2621 


2732 


2843 


2954 


3o64 


3ip 


in 


392 


3286 


3397 


36i8 


3729 


384o 


3950 


4061 


4171 


4282 


in 


3 9 3 


4393 


45o3 


4614 


4724 


4834 


4945 


5o55 


5i65 


5276 


5386 


110 


694 


5496 


56o6 


5717 


582 7 


5 9 3 7 


6047 


6i5 7 


6267 


63 7 7 


6487 


no 


3 9 5 


6597 


6707 
7803 


6817 


6927 


7037 


7146 


7256 


7366 
8465 


7476 


7586 


no 


3 9 6 


7 6 9 5 

8791 

• 9883 


79U 


8024 


^i34 


8243 


8353 


85 7 2 


868 1 


no 


S3S 


8900 


9009 


9119 


9228 


9337 
0428 


9446 


9556 i 9665 


9774 


109 


9992 
1082 


♦ 101 


0210 


o3io 
1408 


o53t 0646 j 0755 


0864 


109 


399 


600973 


1 191 


1299 


1D17 


1620 j 1734 


i843 


1951 


109 


N. 





1 


2 


3 


4 


5 


6 | 7 


8 


9 


D. 



Table I. 


LOGARITHMS OF NUMBERS. 


7 


N. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


D. 


400 


60 2060 


2169 
3253 


2277 


2386 


2494 


26o3 


2711 


2819 


2928 3o36 


108 


401 


3i44 


336i 


3469 


35 77 


3686 


3794 


3902 


4010 


4118 


108 


402 


4226 


4334 


4442 


455o 


4658 


4766 


4874 


4982 


5o8g 


5197 


108 


4o3 


53o5 


54i3 


552i 


5628 


5 7 36 


5844 


5g5i 


6059 
7i33 


6166 


6274 


108 


404 


638 1 


6489 


65 9 6 


6704 


681 1 


69I9 


7026 


7241 


7348 


107 


43f 


7455 


7 562 


7669 


7777 


7884 


7991 


8098 


82o5 


83i2 


8419 

9488 


107 


406 


8526 


8633 


8740 


8847 


8 9 54 


9061 


9167 


9274 


938i 


107 


407 


*9 5 94 


9701 


9808 


9914 


♦021 


0128 


0234 


o34i 


0447 


o554 


107 


408 


61 0660 


0767 


0673 


0979 


1086 


1192 
2254 


1298 


Uo5 


1 5i 1 


1617 


106 


409 


1723 


1829 


1936 


2042 


2148 


236o 


2466 


2572 


2678 


106 


410 


2784 
3842 


2890 


2996 


3l02 


3207 


33i3 


3419 

4475 


3525 


363o 


3736 


106 


4ii 


3 9 47 


4o53 


4159 
52i3 


4264 


4370 


458 1 


4686 


4792 


106 


412 


4897 


Coo3 


5io8 


53i9 


5424 


5529 


5634 


5740 


5845 


io5 


4i3 


SgSo 


6o55 


6160 


6265 


6370 


6476 


658i 


6686 


6790 


68 9 5 


io5 


4i4 


7000 


7io5 


7210 


73i5 


7420 


7025 


7629 


7734 


783 9 


7943 


io5 


4i5 


8048 


8i53 


8 2 5 7 


8362 


8466 


85 7 i 


8676 


8780 


8884 


8989 


io5 


416 


* 9093 


9198 


9302 


9406 


9 5u 


9615 


9719 


9824 


9928 


♦ 032 


104 


417 


62 oi36 


0240 


o344 


0448 


o552 


o656 


0760 


0864 


0968 


1072 


104 


418 


1 1 76 


1280 


1 384 


1488 


1592 


1695 
2732 


283? 


1903 


2007 


2110 


104 


419 


2214 


23i8 


2421 


2525 


2628 


2939 


3o42 


3 146 


104 


420 


3249 


3353 


3456 


3559 


3663 


3 7 66 


386 9 


3973 


4076 


4179 


io3 


421 


4282 


4385 


4488 


4591 


46 9 5 


4798 


4901 


5oo4 


5107 


5210 


io3 


422 


53i2 


541 5 


55i8 


562i 


5724 


582 7 


5929 


6o32 


6i35 


6238 


io3 


423 


634o 


6443 


6546 


6648 


6 7 5i 


6853 


6 9 56 


7o58 


7161 


7263 


io3 


424 


7366 


7468 


7571 


7673 


7775 


7878 


7980 


8082 


8i85 


8287 


102 


425 


838 9 


8491 


85 9 3 


86 9 5 


8797 


8900 


9002 


9104 


9206 


93o8 


102 


426 


*94io 


9512 


g6i3 


9715 


9817 


9919 


♦021 


0123 


0224 


o326 


102 


427 


63 0428 


o53o 


o63i 


0733 


o835 


0936 


io38 


n3 9 


1241 


i342 


102 


428 


1444 


1 545 


1647 


1748 


1849 


1961 


2052 


2 1 53 


2255 


2356 


101 


429 


2457 


2559 


2660 


2761 


2862 


2963 


3o64 


3i65 


3266 


3367 


101 


43o 


3468 


3569 


3670 


3771 


38 7 2 


3973 


4074 


4n5 


4276 


4376 


100 


43 1 


4477 


45 7 8 
5584 


4679 
5685 


4779 
5 7 85 


4880 


4981 


5o8i 


5i82 


5283 


5383 


100 


432 


5484 


5886 


5 9 86 


6087 


6187 


6287 


6388 


100 


433 


6488 


6588 


6688 


6789 


6889 


6989 


7089 


7189 
8190 


7290 


7390 


100 


434 


7490 


7 5 9 o 


7690 


7790 


7890 


7990 


8090 


8290 


838 9 


99 


435 


8489 


858g 


8689 


8789 


8888 


8988 


9088 


9188 


9287 


9387 


99 


436 


*9486 


9586 


9686 


9786 


9 885 


9984 


♦084 


oi83 


0283 


o382 


99 


437 
438 


640481 


o58i 


0680 


0779 


0879 


0978 


1077 


1177 


1276 


i3 7 5 


99 


1474 


i5 7 3 


1672 


1771 


1871 


1970 


2069 
3o58 


2168 


2267 


2366 


99 


439 


2465 


2563 


2662 


2761 


2860 


2959 


3i56 


3255 


3354 


99 


44o 


3453 


355i 


365o 


3749 


3847 


3946 


4044 


4U3 


4242 


434o 


98 


44i 


4439 


4537 


4636 


4734 


4832 


493i 


5029 


5127 


5226 


5324 


98 


442 


5422 


5521 


5619 


5717 


58i5 


5 9 i3 


601 1 


6110 


6208 


63o6 


98 


443 


6404 


65o2 


6600 


6698 


6796 


6894 


6992 


7089 


7187 
8i65 


7285 


98 


444 


7383 


748i 


7579 


7676 


7774 


7872 


7969 


8067 


8262 


98 


445 


836o 


8458 


8555 


8653 


8750 


884 S 


8945 


9043 


9140 


9237 


97 


446 


* 9335 


9432 


953o 


9627 


9724 


9821 


9919 


♦016 


ou3 


0210 


97 


447 


65 o3o8 


o4o5 


o5o2 


0599 


0696 


0793 


0890 


0987 


1084 


1181 


97 


448 


1278 


i375 


1472 


1569 


1666 


1762 


1809 


ig56 


2o53 


2i5o 


97 


449 


2246 


2343 


2440 


2536 


2633 


2730 


2826 


2923 


3019 


3n6 


97 


45o 


32i3 


33oo 
42 7 3 


34o5 


35o2 


35 9 8 


36 9 5 


3791 


3888 


3984 


4080 


96 


45 1 


4177 


4369 


4465 


4562 


4608 


4734 


485o 


4946 


5o42 


06 


452 


5i38 


5235 


533 1 


5427 


55 2 3 


56ig 


5 7 i5 


58io 


5 9 o6 


6002 


96 


453 


6098 
7006 


6194 


6290 


6386 


6482 


6577 


66 7 3 


6769 


6864 


6960 


96 


454 


7i52 


7247 


7343 


7438 


7534 


7629 


7725 


7820 


7916 


96 


455 


801 1 


8107 


8202 


8298 


83 9 3 


8488 


8584 


8679 


8774 


8870 


9 5 


456 


8965 


9060 


9 i55 


9230 


9346 


9441 


9 536 


g63 1 


9726 


9821 


9 5 


45 7 


♦ 9916 


♦on 


0106 


0201 


0296 


0391 


0486 


o58i 


0676 


0771 


9 5 


458 


66 o865 


0960 


io55 


n5o 


1245 


i339 


1434 


1 529 
2475 


1623 


1718 


9 5 


459 


i8i3 


1907 


2002 


2096 


2191 


2286 


238o 


256g 


2663 


9 5 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D." 



8 




LOGARITHMS OF NUMBERS. Table L 


N. 





1 


2 


3 


4 


5 


6 

3324 


7 


8 


9 


D. 


460 


66 2708 


2852 


2947 
388 9 


3o4i 


3i35 


323o 


3418 


35i2 


3607 


94 


461 


3701 


3795 


3 9 83 


4078 


4172 


4266 


436o | 4454 


4548 


94 


4<>2 


4642 


4736 


483o 


4924 


5oi8 


5lI2 


5206 


0299 ' 5393 
6237 j 633i 


5487 


94 


463 


558i 


5675 


5 7 6 ? 


5862 


5956 
6892 


6o5o 


6i43 


6424 


94 


464 


65i8 


6612 


6705 


6799 


6986 


7079 


7173 j 7266 


736o 


94 


465 


7453 

8386 


7546 


7640 


7 7 33 


7826 


7920 


8oi3 


8106' 8199 


8293 


93 


466 


8479 


8572 


8665 


8 7 5 9 


8852 


8945 i 9o38 i 9i3i 


9224 


93 


467 


*93i7 


9410 


95o3 


9596 


9689 


9782 


9875 ; 9967 *o6o 


oi53 


93 


468 


67 0246 


0339 


o43 1 


o524 


0617 


0710 


0802 ! o8g5 0988 


1080 


9 3 


469 


1 173 


1265 


i358 


i45i 


1 543 


1 636 


1728 


1821 


1913 


2oo5 


93 


470 


2098 


2190 


2?83 


23 7 5 


2467 


256o 


2652 


2744 


2836 


2929 
38oo 


92 


471 


3021 


3n3 


3ao5 


3297 


3390 


3482 


3574 


3666 


3 7 58 


92 


472 


3o42 


4o34 


4126 


4218 


43io 


4402 


4494 


4586 


4677 


4769 


92 


473 


4861 


4953 


5o45 


5i37 


5228 


5320 


5412 


55o3 


5595 


5687 


92 


474 


5 77 8 


5870 


5962 


6o53 


6i45 


6236 


6328 


6419 


65u 


6602 


92 


475 


6694 


6 7 85 


6876 


6968 
7881 


7o5 9 


7i5i 
8o63 


7242 
8i54 


7 333 
8245 


7424 


7016 


91 


476 


n 


7698 


7789 
8700 


1972 
8882 


8336 


8427 


91 


477 


8609 


8791 


8 97 3 


9064 


91 55 


9246 


9337 


91 


478 


*9428 


9 5i 9 


9610 


9700 


979 « 


9882 


9973 


♦o63 


01 54 


0245 


91 


479 


68o336 


0426 


o5i7 


0607 


0698 


0789 


0879 


0970 


1060 


ii5i 


91 


480 


1241 


i332 


1422 


i5i3 


i6o3 


1693 


1784 


1874 


1964 


2o55 


90 


481 


2i45 


2235 


2320 


2416 


25o6 


2596 


2686 


2777 


2867 


2957 
385 7 


90 


482 


3o47 


3i3 7 


3227 


33i7 


3407 


3497 


358 7 


36 7 7 


3767 


90 


483 


3 9 47 
4845 


4o37 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


90 


484 


4935 


5o25 


5u4 


5204 


5294 


5383 


5473 


5563 


5652 


90 


485 


5742 


583 1 


5921 


6010 


6100 


6189 

7083 


6279 


6368 


6458 


6547 


89 


486 


6636 


5726 


68i5 


6904 


6994 


7172 
8064 


7261 


735i 


7440 


$ 


487 


7529 


7618 


7707 
85 9 8 
9486 


7796 


7886 


7975 


8i53 


8242 


833 1 


89 


488 


8420 


8509 
9398 


8687 
9 5 7 5 


8776 


8865 


8o53 
9841 


9042 


9 1 3 1 


9220 


89 


489 


*93c>9 


9664 


97 53 


9930 


♦019 


0107 


89 


490 


69 0196 


0285 


0373 


0462 


o55o 


0639 


0728 


0816 


0905 


0993 


81 


491 


1081 


1170 


1258 


1 347 


1435 


i524 


16x2 


1700 


1789 


1877 


492 


io65 
2847 


2o53 


2142 


2230 


23i8 


2406 


2494 


2583 


2671 


2 7 5 9 


88 


493 


2935 
38i5 


3o23 


3 1 1 1 


3i99 
4078 


3287 


3375 


3463 


355i 


363 9 


88 


494 


3727 


3903 


3991 


4166 


4254 


4342 


443o 


45i7 


88 


495 


46o5 


4693 


4781 


4868 


4956 


5o44 


5i3i 


5219 


53o7 


5394 


8S 


496 


5482 


5569 


5657 


5744 


5832 


5 9 i 9 
6793 


6007 


6094 


6182 


6269 


87 


497 


6356 


6444 


653 1 


6618 


6706 


6880 


6968 


7o55 


7142 
8014 


* 7 


498 


7229 


7 3i 7 
8188 


74o4 


7491 


7 5 7 8 


7665 


7 7 5a 


7839 


7926 


? 7 


499 


8101 


8273 


8362 


8449 


8535 


8622 


8709 


8796 


8883 


87 


5oo 


8070 
* 9 838 


9 o5 7 


9U4 


923i 


9 3i 7 


9404 


9491 


9 5 7 8 


9664 


975i 


S 7 


5oi 


9924 


♦on 


0098 


0184 


0271 


o358 


0444 


o53i 


0617 


87 


502 


70 0704 


0790 


0877 


oo63 


io5o 


u36 


1222 


1 309 


1395 


1482 


86 


5o3 


1 568 


1604 


1741 


1827 


1913 


1999 


20S6 


2172 


2238 


2344 


86 


5o4 


243 1 


2517 


26o3 


2689 


2773 


2861 


2947 


3o33 


3i 19 


32o5 


86 


5o5 


3291 


3377 


3463 


354Q 
4408 


3635 


3721 


3807 


38 ? 5 


3979 


4o65 


86 


5o6 


4i5i 


4236 


4322 


4494 


4579 


4665 


47^1 


4S3 7 


4922 


86 


507 


5oo8 


5094 


5179 


5265 


53do 


5436 


5522 | 0607 


56 9 3 


5 77 8 


86 


5o8 


5864 


5949 


6o33 


6120 


6206 


6291 


6376 | 6462 


6547 


6632 


85 


509 


6718 


68o3 


6888 


6974 


7009 


7144 


7229 J 73 1 5 


7400 


7485 


85 


5io 


7570 


7 655 


7740 


7826 


791 1 


7996 

8846 


8081 J 8166 


825i 


8336 


85 


5n 


8421 


85o6 


85gi 


8676 


8761 


8931 J 9015 


c 1 a 


9i85 


85 


512 


* 9270 


9355 


944o 


9524 


9609 


9694 


9779 9S63 


9948 


♦o33 


85 


5i3 


710117 


0202 


0287 


0371 


0456 


0340 


0625 0710 


0794 
1639 


0879 
1723 


85 


5i4 


0963 


1048 


n3a 


1217 


i3oi 


i385 


1470 


1334 


84 


5i5 


1807 


1892 


1976 


2060 


2144 


2229 


23i3 


2397 


2481 


2566 


84 


5i6 


265o 


2734 


2818 


2902 


2986 


3070 


3i54 


3238 


3323 


3407 


84 


5ii 


34QI 
433o 


3575 


365o 


3742 


3826 


3910 


3994 
4833 


4078 


4162 


4246 


84 


5i8 


4414 


44Q7 
5335 


458i 


4665 


4749 
55S6 


4916 


5ooo 


5o34 


84 


5lQ 


5167 


525i 


0418 


55o2 


5669 5753 


5836 


3920 


84 


N. 





1 





3 


* 


5 


6 i 7 


8 


9 


D. 



Table I. 


LOGARITHMS OF NUMBERS. 


9 


N. 





1 


2 


3 1 4 


5 


6 


7 


8 


9 


D. 


520 


71 6oo3 


6087 


6170 


6254 


6337 


6421 


65o4 


6588 


6671 


6 7 54 


83 


521 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


75o4 


7 58 7 


83 


522 


7671 


7754 
8585 


7 83 7 


7920 


8oo3 


8086 


8169 


8253 


8336 


8419 
9248 


83 


523 


85o2 


8668 


8 7 5i 


8834 


8917 


9000 


9083 


9i65 


83 


524 


* g33i 


94U 


9497 


9380 


9 663 


9745 


9828 


991 1 


9994 


♦077 


83 


525 


72 0159 


0242 


o325 


1233 


0490 


0573 


o655 


0738 


0821 


0903 


83 


526 


0986 
1811 


1068 


1 1 5 1 


i3i6 


i3 9 8 


1481 


1 563 


1646 


1728 


82 


527 
528 


1893 


1975 


2o58 


2140 


2222 


23o5 


238 7 


2469 


2552 


82 


2634 


2716 


2798 


2881 


2963 


3o45 


3127 


3209 


3291 


3374 


82 


529 


3456 


3538 


3620 


3702 


3784 


3866 


3948 


4o3o 


4112 


4194 


82 


53o 


4276 


4358 


444o 


4522 


4604 


4685 


4767 


4849 


493i 


5oi3 


82 


53 1 


5095 


5i 7 6 


5258 


534o 


5422 


55o3 


5585 


566 7 


5748 


583o 


82 


532 


5912 


5993 
6809 
7623 


6075 


6i56 


6238 


6320 


6401 


6483 


6564 


6646 


82 


533 


6727 


6890 


6972 


7o53 


7i34 


7216 


7297 
8110 


7 3 79 


746o 
8273 


81 


534 


7541 


7704 


77 85 


7866 


7948 


8029 


8191 


81 


535 


8354 


8435 


85i6 


85 9 7 


8678 


8 7 5 9 


8841 


8922 


9003 


9084 


81 


536 


9165 


9246 


9 32 7 


9408 


948o 


9 5 7 o 


965i 


97 32 


9813 


9893 


81 


53 7 


*9974 


♦o55 


oi36 


0217 


0298 


o3 7 8 


0459 


o54o 


0621 


0702 


81 


538 


730782 


o863 


0944 


1024 


uo5 


1 186 


1266 


1 347 


1428 


i5o8 


81 


53 9 


1 589 


1669 


ipo 


i83o 


1911 


1991 


2072 


2l52 


2233 


23i3 


81 


54o 


23 9 4 


2474 


2555 


2635 


2715 


2796 


2876 


2956 


3o37 


3i 1 7 


80 


541 


3i 97 


32 7 8 


3358 


3438 


35i8 


35 9 8 


3679 
4480 


3]5 9 


383 9 


3 9 i 9 


80 


542 


3999 
4800 


4079 


4160 


4240 


4320 


44oo 


456o 


4640 


4720 


80 


543 


4880 


4960 


5o4o 


5 1 20 


5200 


5279 


535 9 


5439 


5519 


80 


544 


5599 


5679 


5 7 5 9 


5838 


5 9 i8 


5998 


6078 


6i5 7 


6237 


63i 7 


80 


545 


63 97 


6476 


6556 


6635 


6715 


6795 


6874 


6 9 54 


7034 


71 13 


80 


546 


7193 
7987 


7272 


7352 


143 1 
822D 


7611 


7590 
8384 


7670 
8463 


7749 
8543 


7829 


7908 


79 


547 
548 


8067 


8146 


83o5 


8622 


8701 


79 


8781 


8860 


8939 


9018 


9097 


9177 


9256 


9 335 


94i4 


9493 


79 


549 


#9572 


9651 


97 3 1 


9810 


9889 


9968 


♦047 


0126 


0205 


0284 


79 


55o 


74 o363 


0442 


0521 


0600 


0678 


0757 


o836 


0915 


0994 

1782 


1073 


79 


55i 


Il52 


1230 


1 309 


i388 


1467 


1546 


1624 


1703 


i860 


79 


552 


1939 


2018 


2096 


2175 


2254 


2332 


241 1 


2489 
3275 


2568 


2646 


79 


553 


2723 


2804 


2882 


2961 


3o39 
3823 . 


3n8 


3196 


3353 


343 1 


7 8 


554 


35io 


3588 


3667 


3745 


3902 


3 9 8o 


4o58 


4i36 


42i5 


78 


555 


4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 


78 


556 


5075 


5i53 


523i 


5309 


538 7 


5465 


5543 


562i 


5699 


5777 


78 


55i 

558 


5855 


5 9 33 


6011 


6089 


6167 


6245 


6323 


6401 


6479 


6556 


78 


6634 


6712 


6 8° 

7 56 7 


6868 


6945 


7023 


7101 


7179 
79 d5 


7256 
8o33 


7334 


78 


55 9 


7412 


7489 


7645 


7722 


7800 


7878 


8110 


78 


56o 


8188 


8266 


8343 


8421 


8498 


8576 


8653 


8 7 3i 


8808 


8885 


77 


56 1 


8 9 63 


9040 


9118 


9 i 9 5 


9272 


935o 


9427 


9304 


9 582 


9659 


77 


562 


#9736 


9814 


9891 


9968 


♦o45 


OI23 


0200 


0277 
1048 


o354 


o43 1 


77 


563 


75 o5o8 


o586 


o663 


0740 


0817 


0894 


0971 


1 125 


1202 


'77 


564 


1279 


i356 


1433 


i5io 


1 58 7 


1664 


1741 


1818 


1895 


1972 


77 


565 


2048 


2125 


2202 


2279 


2356 


2433 


2 509 


2586 


2663 


2740 


77 


566 


2816 


2893 


2970 


3o47 


3i23 


3200 


3277 


3353 


343o 


35o6 


77 


567 


3583 


366o 


3 7 36 


38i3 


388 9 


3966 


4042 


4119 

4883 


4195 


4272 


77 


568 


4348 


4425 


45oi 


4578 


4654 


4730 


4807 


4960 


5o36 


76 


569 


5lI2 


5i8 9 


5265 


534i 


5417 


5494 


5570 


5646 


5722 


5799 


76 


5io 


58 7 5 


5951 


6027 


6io3 


6180 


6256 


6332 


6408 


6484 


656o 


76 


5 7 i 


6636 


6712 


6788 
7548 


6864 


6940 


7016 


7092 


7168 


7244 


7320 


76 


572 


7396 
8i55 


7472 


7624 


7700 


7775 


785i 
8609 


7927 
8685 


8oo3 


8079 


76 


5 7 3 


823o 


83o6 


8382 


8408 


8533 


8761 


8836 


76 


574 


8912 


8988 


9063 


9 i3 9 


9214 


9290 


9366 


9441 


9 5i 7 


9592 


76 


5 7 5 


#9668 


9743 


9819 


9894 


9970 


♦o45 


0121 


0196 


0272 


o347 


75 


5 7 6 


760422 


0498 

1231 


0573 


0649 


0724 


°Tl 9 

i552 


0875 


0960 


1025 


IIOI 


75 


m 


1 176 


i326 


1402 


•477 
2228 


1627 


1702 


1778 


1 853 


75 


1928 


2003 


2078 


2i53 


23o3 


23 7 8 


2453 


2529 

3278 


2604 


75 


5 79 


2679 


2754 


2829 


2904 


2978 


3o53 


3i28 


32o3 


3353 


75 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



10 




LOGARITHMS OF NUMBERS. Table I. 


N. 





1 


2 


3 


4 


5 1 6 


7 


8 


9 


D. 


58o 


"6 3428 


35o3 


35 7 8 


3653 


3727 


38o2 3877 


3952 


4027 


4101 


75 


58 1 


4176 


425i 


4326 


4400 


4475 


455o 


4624 


4699 


4774 


4848 


75 


58a 


4923 


4998 


5072 


5i47 


5221 


5296 


5370 


5445 


5520 


55 9 4 
6338 


75 


583 


566 9 
641 3 


5 7 43 


58i8 


58 9 2 
6636 


5 9 66 


6041 


6u5 


6190 
6 9 33 


6264 


74 


584 


6487 


6562 


671O 


6 7 85 


685 9 


7007 


7082 


74 


585 


7 i56 


723o 


73o4 


7379 


7453 
8l 9 4 


7527 
8268 


7601 


7675 


7749 
8490 


7823 


74 


586 


7898 


7972 


8046 


8120 


8342 


8416 


8564 


74 


58 7 
588 


8638 


8712 


8786 


8860 


8 9 34 


9008 


9082 


9i56 


923o 


g3o3 


74 


* 9 3 7 7 
77 ou5 


945 1 


9525 


9 5 99 


9673 


9746 


9820 


9894 


9968 


♦042 


74 


58 9 


0189 


0263 


o336 


0410 


0484 


o557 


o63i 


0705 


0778 


74 


590 


o852 


0926 


0999 


1073 


1 146 


1220 


1293 


i36 7 


1440 


i5i4 


74 


591 


1587 


1661 


1734 


1808 


l88l 


1955 


2028 


2102 


2175 


2248 


73 


592 


2322 


2395 


2468 


2542 


26i5 


2688 


2762 


2835 


2908 


2981 


73 


5 9 3 


3o55 


3i28 


3201 


3274 


3348 


3421 


3494 


356 7 


3640 


3 7 i3 


73 


594 


3786 


386o 


3 9 33 


4006 


4079 


4i52 


4223 


4298 


4371 


4444 


73 


5 9 5 


45i 7 


4590 


4663 


4736 


4809 
5538 


4882 


4955 


5028 


5ioo 


5i 7 3 


73 


5 9 6 


5246 


53i9 


5392 


5465 


56io 


5683 


5736 


582 9 


5902 


73 


III 


5974 


6047 


6120 


6193 


6265 


6338 


641 1 • 


6483 


6556 


6629 


7 ? 


6701 


6774 


6846 


6919 


6992 


7064 


7137 


7209 


7282 


7354 


73 


5 99 


7427 


7499 


7 5 7 2 


7644 


7717 


7789 


7862 


7 9 34 


8006 


8079 


72 


600 


8i5i 


8224 


8296. 


8368 


844i 


85i3 


8585 


8658 


8730 


8802 


72 


601 


8874 


8947 


9019 


9°9* 


9163 


9236 


93o8 


9 38o 


9452 


9524 


72 


602 


#9596 


9669 


974i 


9813 


9 885 


9957 


♦029 


0101 


0173 


0245 


72 


6o3 


780317 


o38 9 


0461 


o533 


o6o5 


0677 


0749 
1468 


0821 


o8 9 3 


0965 


72 


604 


io37 


1 109 


1181 


1253 


1324 


1396 


1 54o 


1612 


1684 


72 


6o5 


1755 


1827 


1899 


1971 


2042 


2H4 


2186 


2258 


2329 


2401 


7* 


606 


247 3 


2344 


2616 


2688 


2 7 5o 
3475 


283i 


2902 


2974 


3o46 


3i 17 


7 } 


607 


3i8 9 


3260 


3332 


34o3 


3546 


36i8 


3689 


3761 


3832 


7« 


608 


3904 


3975 


4046 


41 18 


4189 


4261 


4332 


44o3 


4475 
5i8 7 


4546 


7i 


609 


4617 


4689 


4760 


483 1 


4902 


4974 


5o45 


5n6 


5259 


71 


610 


533o 


5401 


5472 


5543 


56i5 


5686 


5 7 5 7 


5828 


58 9 g 


5970 


71 


611 


6o4i 


6112 


6i83 


6 2 54 


6325 


63 9 6 


6467 


5538 


6609 


6680 


7» 


612 


6751 


6822 


68 9 3 


6964 


7o35 


7106 


7177 7248 


73i 9 


7390 


7i 


6i3 


7460 
8168 


753 1 


7602 


7673 


7744 


7815 


7885 


79 36 


8027 


8098 


7i 


614 


8239 


83io 


838i 


845i 


8522 


85 9 3 


8663 


8734 


8804 


li 


6i5 


8875 


8946 


9016 


9087 


9157 


9228 


9299 


9369 


9440 


95io 


7i 


616 


* 9 58i 


965 1 


9722 


9792 


9863 


9933 


♦004 


0074 


0144 


02 1 5 


70 


617 
618 


79 0285 


o356 


0426 


0496 


0567 


0637 


0707 


0778 


0848 


0918 


70 


0988 


1009 


1129 


1 199 


1269 


i34o 


1410 


1480 


i55o 


1620 


70 


619 


1691 


1761 


i83i 


1901 


1971 


2041 


21 1 1 


2181 


2232 


2322 


70 


620 


2392 


2462 


2532 


2602 


2672 


2742 


2812 


2882 


2952 


3022 


70 


621 


3092 


3i62 


3a3i 


33oi 


3371 


344i 


35n 


358i 


365i 


3721 


70 


622 


3790 


386o 


3930 


4000 


4070 


4139 


4209 


4279 


4349 


44l8 


70 


623 


4488 


4558 


4627 


4697 


4707 


4836 


4906 


4976 


5o43 


5n5 


70 


624 


5i85 


5234 


5324 


53 9 3 


5463 


5532 


56o2 


5672 


5741 


58n 


70 


625 


588o 


5 9 49 


6019 
6713 


6088 


6i58 


6227 


6297 


6366 


6436 


65o5 


69 


626 


65 7 4 


6644 


6782 


6852 


6921 


6990 
7683 
83 7 4 


7060 


7129 


7198 


69 


627 


7268 


7 33 7 


74o6 


7475 


7545 


7614 


77 52 

8443 


7821 


7800 


69 


628 


7960 
865 1 


8029 


8098 


8167 


8236 


83o5 


85i3 


8582 


69 


629 


8720 


8789 


8858 


8927 


8996 


9065 


9i34 


9203 


9272 


69 


63o 


9341 


9409 


9478 


9 547 


9616 


9 685 


97 54 


9823 


9802 


9961 


5 9 


63 1 


bo 0029 


0098 
0786 


0167 


0236 


o3o5 


o373 


0442 


031 1 


o58o 


0648 


69 


632 


0717 


o854 


0923 


0992 


1061 


J o 2 2 


II98 

l884 


1266 


i335 


69 


633 


i4o4 


1472 


i54i 


1609 


1678 


1747 


i8i5 


1952 


2021 


69 


634 


2089 


2i58 


2226 


2293 


2363 


2432 


2300 


2568 


2637 


2703 


69 


635 


2774 


2842 


2910 


2979 


3o47 


3n6 


3i84 


3252 


3321 


3389 


68 


636 


345 7 


3523 


3594 


3662 


373o 


3798 


3867 | 3 9 35 
4548 4616 


4oo3 


4071 


68 


637 
638 


4i3 9 


4208 


4276 


4344 


44i2 


4480 


4685 


4753 


68 


4821 


4889 


4957 


5o25 


5093 


! 5i6i 


522o 5297 

5908 1 5976 


5365 


5433 


68 


63 9 


55oi 


5569 


5637 


5705 


5 77 3 


J 5841 


6o44 


6112 


68 
D. 


N. 





1 


2 | 3 | 4 


1 « 


6 | 7 


8 


9 



Table I. LOGARITHMS OF NUMBERS. 11 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


640 


806180 


6248 


63i6 


6384 


645 1 


6519 


6587 


6655 


6723 


6790 


68 


641 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7 332 
8008 


7400 
8076 


7467 


68 


642 


7 535 


7603 

8279 
8 9 53 


7670 
8346 


7738 


-7806 


7873 
8549 


794i 


8i43 


68 


643 


8211 


8414 


8481 


8616 


8684 


8 7 5i 


8818 


67 


644 


8886 


9021 


9088 


9i56 


9223 


9290 


9 358 


9425 


9492 


67 


645 


*o56o 


9627 


9694 


9762 


9829 


9896 


9964; 


♦o3i 


0098 


oi65 


67 


646 


81 0233 


o3oo 


0367 


0434 


o5oi 


0569 


o636 


0703 


0770 


0837 
i5o8 


67 


647 


0904 


0971 


1039 


1 106 


n 7 3 


1240 


1 307 


1374 


i44i 


67 


648 


i5 7 5 


1642 


1709 


1776 


i843 


1910 


1977 


2044 


21 1 1 


2178 


67 


649 


2245 


23l2 


23 79 


2445 


25l2 


2379 


2646 


2713 


2780 


2847 


67 


65o 


2913 


2980 


3o47 


3n4 


3i8i 


3247 


33i4 


338i 


3448 


35i4 


67 


65i 


358i 


3648 


37i4 


3781 


3848 


3914 


3981 


4048 


4u4 


4181 


67 


652 


4248 


43 14 


438 1 


4447 
5u3 


45i4 


458 1 


4647 


47i4 


4780 


4847 


67 


653 


49i3 


4980 


5o46 


5179 

5843 


5246 


53i2 


53 7 8 


5445 


55n 


66 


654 


55 7 8 


5644 


5711 


5777 


5910 


5976 


6042 


6109 


6i 7 5 


66 


.655 


6241 


63o8 


63 7 4 


6440 


65o6 


65 7 3 


663g 


6705 


6771 


6838 


66 


656 


6904 


6970 
t63i 
8292 
8 9 5i 


7o36 


7102 


7169 


7235 


73oi 


8028 


7433 
8094 
8754 


7499 
8160 


66 


65 7 


7565 
8226 


his 


7764 


783o 


7896 
8556 


7962 


66 


658 


8424 


8490 


8622 


8688 


8820 


66 


65 9 


8885 


9017 


9083 


9149 


9215 


9281 


9346 


9412 


9478 


66 


660 


#9544 


9610 


9676 

o333 


974i 


9807 


9873 


I2DI 


♦004 


0070 


oi36 


66 


661 


82 0201 


0267 


0399 
io55 


0464 


o53o 


0661 


0727 


0792 


66 


662 


o858 


0924 
1 5 79 


0989 
i645 


1 1 20 


1 186 


1317 


i382 


1448 


66 


663 


i5i4 


1710 


1775 


1841 


IO06 

2060 


1972 
2626 


2037 


2103 


65 


664 


2168 


2233 


2299 


2364 


243o 


2495 


2691 


2756 


65 


665 


2822 


2887 


2952 


3oi8 


3o83 


3i48 


32i3 


3279 


3344 


3409 


65 


666 


3474 


3539 


36o5 


3670 


3735 


38oo 


3865 


3930 
458i 


3996 


4061 


65 


667 
668 


4126 


4191 


4256 


4321 


4386 


445 1 


45i6 


4646 


471 1 


65 


4776 


4841 


4906 


4971 


5o36 


5ioi 


5i66 


523i 


5296 


536 1 


65 


£69 


5426 


5491 


5556 


5621 


5686 


6751 


58i5 


588o 


5 9 45 


6010 


65 


670 


6075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


65 9 3 


6658 


65 


671 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


73o5 


65 


672 


7369 
8oi5 


7434 


7499 
8i44 


7563 


7628 


7692 
8338 


7757 


7821 


7886 
853 1 


79 5i 


65 


673 


8080 


8209 
8853 


8273 


8402 


8467 


85o5 
9239 


64 


674 


8660 


8724 


8789 


8918 


8982 


9046 


91 1 1 


9175 


64 


6 7 5 


93o4 


9 368 


9432 


9497 
0139 


956 1 


9625 


9690 


9754 


9818 


9882 


64 


676 


*9947 


♦on 


0075 


0204 


0268 


o332 


0396 


0460 


o525 


64 


677 


83 o58 9 


o653 


0717 
i358 


0781 


0845 


0909 
i55o 


o 97 3 


1037 
1678 


1 1 02 


1 166 


64 


678 


1230 


1294 


1422 


i486 


1614 


1742 


1806 


64 


679 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


238i 


2445 


64 


680 


2509 


25 7 3 


2637 
32 7 5 


2700 


2764 


2828 


2892 
353o 


2956 
35 9 3 
423o 


3020 


3o83 


64 


681 


3i47 


321 1 


3338 


3402 


3466 


365 7 


3721 


64 


682 


3 7 84 


3848 


3oi2 


3975 


4039 


4io3 


4166 


4294 


4357 
4993 


64 


683 


4421 


4484 


4548 


461 1 


46 7 5 


4739 
53 7 3 


4802 


4866 


4929 


64 


684 


5o56 


5l20 


5i83 


5247 


53io 


5437 


55oo 


5564 


5627 


63 


685 


5691 


5754 


5817 


588 1 


5944 


6007 


6071 


6i34 


6197 
683o 


6261 


63 


686 


6324 


6387 


645i 


65i4 


65 77 


6641 


6704 


6767 
7399 
8o3o 


6894 


63 


687 


b tl 


7020 


7083 


7U6 


7210 


7273 


7336 


7462 
8o 9 3 


7525 


63 


688 


7588 
8219 


7652 


77'5 


7778 


7841 


7904 


7967 
85 9 7 


8i56 


63 


689 


8282 


8345 


8408 


8471 


8534 


8660 


8723 


8786 


63 


6 9 c 


8849 
*9478 


8912 
9^41 


8 97 5 


9038 


9101 


9164 


9227 

9 855 


9289 
9918 


9352 


941 5 


63 


691 


9604 


9667 


9729 


9792 


9981 


♦o43 


63 


692 


84 0106 


0169 


0232 


0294 


o357 


0420 


0482 


o545 


0608 


0671 


63 


6 9 3 


0733 


0796 


o85 9 
1485 


0921 
1 547 


0984 


1046 


1109 
1735 


1172 


1234 


1297 


63 


694 


i359 


1422 


1610 


1672 


1797 


i860 


1922 


63 


6 9 5 


i 9 85 


2047 


2110 


2172 


2235 


2297 


236o 


2422 


2484 


2547 


62 


696 


2609 
3233 


2672 


2734 


2796 


285 9 


2921 


2983 


3o46 


3io8 


3170 


62 


697 


3295 


335 7 


3420 


3482 


3544 


36o6 


366 9 


3 7 3i 


3793 


62 


698 


3855 


3oi8 
453 9 


3980 


4042 


4104 


4166 


4229 


4291 


4353 


44i 5 


62 


699 


4477 


4601 


4664 


4726 


4788 


485o 


4912 


4974 


5o36 


62 
1). 


N. 





1 


2 


3 


4 


r * 


6 


7 


8 


9 



12 




LOGARITHMS OF NUMBERS. Table L 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


700 


845098 


5i6o 


5222 


5284 


5346 


5408 


5470 


5532 


5594 


5656 


62 


701 


5 7 i8 


5780 


5842 


5904 


5 9 66 


6028 


6090 


6i5i 


62i3 


6275 


62 


702 


6337 


53 99 


6461 


6323 


6385 


6646 


6708 


6770 


6832 


6894 


62 


7o3 


6955 
7 5 7 3 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 
8066 


75n 
8128 


62 


704 


7634 


7696 


77 58 


7819 


7881 


7943 


8004 


62 


7o5 


8189 


825i 


83i2 


83 7 4 


8a35 


8497 


855g 


862c 


8682 


8743 


62 


706 


88o3 


8866 


8q28 


8989 


9o5i 


9112 


9I 7i 


9235 


9297 


9 3 58 


61 


708 


9419 

85oo33 


9481 


9342 


9604 


9663 


9726 


9788 


9849 


991 1 


997? 


61 


0095 


oi56 


0217 


0279 


o34o 


0401 


0462 


0324 


0383 


61 


709 


0646 


0707 


0769 


o83o 


0891 


0952 


1014 


1073 


n36 


1197 


61 


710 


1258 


l320 


i38i 


1442 


•5o3 


1 564 


1625 


1686 


1747 


1809 


61 


711 


1870 


1931 


1992 


2o53 


2114 


2175 


2236 


2297 


2358 


2419 


61 


712 


2480 


2341 


2602 


2663 


2724 


2783 


2846 


2907 


2968 


3029 


61 


7 i3 


3090 


3i5o 


3211 


3272 


3333 


33 9 4 


3455 


35i6 


3377 


3637 


61 


714 


36 9 8 


3 7 5 9 


3820 


388i 


3941 


4002 


4o63 


4124 


4i85 


4245 


61 


7 i5 


43o6 


4367 


4428 


4488 


4549 


4610 


4670 


473i 


4792 


4852 


61 


716 


49i3 


4974 


5o34 


5095 


5 1 56 


52i6 


5277 


533 7 


53 9 8 


5459 


61 


7*7 


5519 


558o 


564o 


5701 


5761 


5822 


5882 


5943 


6oo3 


6064 


61 


718 


6124 


6i85 


6245 


63o6 


6366 


6427 


6487 


6348 


6608 


6668 


60 


719 


6729 


6789 


685o 


6910 


6970 


7o3i 


7091 


7i52 


7212 


7272 


60 


720 


7332 


7 3 9 3 


7453 
8o56 


75i3 


7574 


7634 


7694 


77 55 


7 8i5 

8417 
9018 


7875 
8477 


60 


721 


7 o35 


7990 


8116 


8176 


8236 


8297 


8357 


60 


722 


8o3 7 


8097 


865 7 


8718 


8778 


8838 


8898 


8 ? 58 


9078 


60 


7 23 


9 ,38 


9198 


9208 


9318 


9 3 79 


9439 


9499 
0098 


9359 


9610 
0218 


9679 
0278 


60 


724 


*97 3 9 


9799 


9859 


9918 


9978 


♦o38 


0138 


60 


725 


86o338 


0398 


0458 


o5i8 


0578 


0637 


0697 


0757 


0817 


0877 


60 


726 


0937 


0996 


io56 


1 1 16 


1 176 


1236 


1293 


i355 


I4i5 


1475 


60 


$ 


1 534 


1094 


i654 


1714 


1773 


i833 


i8 9 3 


1932 


2012 


2072 


60 


2l3l 


2191 


225l 


23io 


2370 


243o 


2489 


2349 


2608 


2668 


60 


729 


2728 


2787 


2847 


2906 


2966 


3o25 


3o83 


3i44 


3204 


3263 


60 


73o 


3323 


3382 


3442 


35oi 


356i 


3620 


368o 


3739 


3799 


3858 


5 9 


7 3i 


3917 


3977 


4o36 


4096 


4i55 


4214 


4274 


4333 


4392 


4452 


59 


7 32 


45u 


4570 


463o 


4689 


4748 


4808 


4867 


4926 


4983 


5o45 


5o 


733 


5io4 


5i63 


5222 


5282 


534i 


54oo 


5439 


5319 


5578 


5637 


5 9 


734 


56 9 6 


5760 


58i4 


58 7 4 


5 9 33 


5992 


6o5i 


6no 


6169 


6228 


5 9 


7 35 


6287 
6878 


6346 


64o5 


6465 


6524 


6583 


6642 


6701 


6760 


6819 


5 9 


736 


6 § 3j 


6996 
7385 


7055 


7"4 


7173 


7232 


7291 


735o 


7409 


i 9 


737 


7467 


7326 


7644 
8233 


7703 


7762 


7821 


7880 


79 3 9 


7908 


29 


738 


8o56 


8n5 


8174 


8292 


835o 


8409 


8468 


8327 


8586 


5 9 


7 3 9 


8644 


8703 


8762 


8821 


8879 


8 9 38 


8997 


9036 


9114 


9173 


5 9 


74o 


923a 


9290 


9349 


9408 


9466 


9523 


9 584 


9642 


9701 


9760 


5 9 


74i 


*98i8 


9877 


9933 


9994 


♦o53 


on 1 


0170 


0228 


0287 


o345 


U 


742 


87 0404 


0462 


0521 


0379 


o638 


0696 


0755 


o8i3 


0872 


0930 


743 


0980 
1D73 


1047 


1 106 


1 164 


1223 


1281 


i33o 
192J 


1398 
1981 


1456 


i5i5 


58 


744 


i63i 


1690 


1748 


1806 


i865 


2040 


2098 


58 


745 


2i56 


22l5 


2273 


233i 


238g 


2448 


25o6 


2564 


2622 


2681 


58 


746 


2739 


2797 


2855 


2913 


2972 


3o3o 


3o88 


3 146 


3204 


3262 


58 


747 


332i 


3379 


3437 


3495 


3353 


36n 


366 9 


3727 


3 7 85 


3844 


58 


748 


3902 


3960 


4018 


4076 


4i34 


4192 


425o 


43o8 


4366 


4424 


58 


749 


4482 


4040 


4598 


4656 


47U 


4772 


483o 


4888 


4945 


5oo3 


58 


75o 


5o6i 


5i 19 


5177 


5235 


5293 


535i 


5409 


5466 


5324 


5582 


58 


7 5i 


564o 


5698 


5 7 56 


58i3 


58 7 i 


5929 


5987 ! 6045 


6102 


6160 


58 


752 


6218 


6276 


6333 


6391 


6449 


6307 


6564 1 6622 


6680 


6737 


58 


7 53 


6795 


6853 


6910 


6968 


7026 


7083 


7141 7199 


7256 


73i4 


58 


754 


7371 


7429 


7487 


7344 


7602 


7 65 9 


7717 7774 


7832 


7889 


58 


755 


7947 

8322 


8004 


8062 


8119 


8177 


8234 


8292 1 8349 


8407 


8464 


57 


7 56 


8579 


8637 


86q4 


8732 


8809 
9 383 


8866 8924 


8981 


9039 


57 


n 


9096 


9i53 


9211 


9268 


9325 


9440 J 9497 


9355 


9612 


57 


♦ 9669 


9726 


9784 


9841 


9898 


9956 


♦01 3 j 0070 


0127 


oi85 


57 


7 5 9 


88 0242 


0299 


o356 


041 3 


0471 


o528 


o585 j 0642 


0699 


0756 


57 


N. 





1 


2 


3 


4 


5 


6 | 7 


8 


9 


D. 



Table I. 


LOGARITHMS OF NUMBERS. 18 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


760 


880814 


0871 


0928 


0985 


1042 


1099 


u56 


I2l3 


1271 


i328 


~5t" 


761 


i385 


1442 


1499 


i656 


i6i3 


1670 


1727 


1784 


1841 


1898 


5? 


762 


iq55 

2325 


2012 


2069 
2638 


2126 


2i83 


2?4o 


2297 


2354 


241 1 


2468 


57 


763 


258i 


2695 


?752 


2809 


2866 


2923 


2980 


3o37 


5 7 


764 


3093 


3i5o 


3207 


3264 


3321 


3377 


3434 


3491 


3548 


36o5 


57 


765 


366 1 


3 7 i8 


3775 
4342 


383a 


3888 


3945 

4612 


4002 


4059 

4625 


4n5 


4172 


57 


766 


4229 

4795 


4285 


4399 


4455 


4569 


4682 


473q 
53o5 


57 


767 


4852 


4909 


496D 


5022 


5078 


5i35 


5192 

5 7 5 7 


5248 


57 


768 


536i 


54i8 


5474 


553 1 


558 7 


5644 


5700 


58i3 


58 7 o 


U 


769 


5926 


5 9 83 


6039 


6096 


6i52 


6209 


6265 


632i 


6378 


6434 


770 


6491 


6547 


6604 


6660 


6716 


6 77 3 


6829 


6885 


6942 

75o5 


6998 


56 


77i 


7004 


7111 


7167 


7223 


7280 


7336 


7 9 55 


7449 
801 1 


756i 


56 


772 


7 6l 7 
8179 


7674 


773o 


7786 


7842 
8404 


7898 


8067 


8i23 


56 


773 


8236 


8292 
8853 


8348 


8460 


85i6 


85 7 3 


8629 


8685 


56 


774 


8741 


8797 


8909 


8 9 65 


9021 


9077 


9134 


9190 


9246 


56 


775 


9302 


9 358 


9414 


9470 


9526 


9582 


9 638 


9694 
0253 


975o 


9806 


56 


776 


*9862 


9918 


9974 


♦o3o 


0086 


0141 


0197 


0309 


o365 


56 


777 
778 


890421 


o477 
io35 


o533 


o58 9 


0645 


0700 


0706 


0812 


0868 


0924 


56 


0980 
1 53 7 


1091 


1147 


1203 


1259 


i3i4 


1370 


1426 


1482 


56 


779 


1593 


1649 


1706 


1760 


1816 


1872 


1928 


i 9 83 


2039 


56 


780 


2095 


2i5o 


2206 


2262 


23i7 


23 7 3 


2429 


2484 


254o 


2595 


56 


781 


265i 


2707 


2762 


2818 


2873 


2929 


2985 
354o 


3o4o 


3096 
365i 


3ioi 


56 


782 


3207 


3262 


33i8 


33 7 3 


3429 


3484 


35o5 
4i5o 


3706 


56 


7 83 


3762 


3817 


38 7 3 


3928 


3984 
4538 


4039 
4593 


4094 


42o5 


4261 


55 


784 


43i6 


437i 


4427 


4482 


4648 


4704 


4759 


4814 


55 


785 


4870 


4925 


4980 
5533 


5o36 


5091 


5 146 


5201 


525 7 


53i2 


5367 


55 


786 


5423 


5478 


5588 


5644 


5699 
6261 


5754 


58o 9 


5864 


5920 


55 


787 


5q7D 


6o3o 


6o85 


6140 


6i 9 5 


63o6 


636 1 


6416 


6471 


55 


788 


6626 


658i 


6636 


6692 


6747 


6802 


685 7 


6912 


6967 
75i 7 


7022 


55 


789 


7077 


7i32 


7187 


7242 


7297 


7352 


7407 


7462 


7572 


55 


790 


7627 


7682 


7737 


7792 


7847 


1902 
845i 


7 9 5 7 
85o6 


8012 


8067 
86i5 


8122 


55 


791 


8176 


823i 


8286 


834i 


83 9 6 


856i 


8670 


55 


792 


8725 


8780 


8835 


8890 


8944 


8099 
9547 


9054 


9109 


9164 


9218 


55 


79 3 


9273 


9328 


9 383 


94J7 
9985 


9492 


9602 


9 656 


9711 


9766 


55 


794 


#9821 


9875 


993o 


♦039 


0094 


0149 


0203 


0258 


03l2 


55 


79 5 


90 0367 


0422 


0476 


o53i 


o586 


0640 


0695 


0749 

1295 


0804 


0859 


55 


7q6 


0913 


0968 


1022 


1077 


u3i 


1186 


1240 


1 349 


1404 


55 


797 
798 


1458 


i5i3 


1 567 


1622 


1676 


1731 


i 7 85 


1840 


1894 
2438 


1948 


54 


2003 


2057 


2112 


2166 


2221 


2275 


2329 
287J 


2384 


2492 

3o36 


54 


799 


2547 


2601 


2655 


2710 


2764 


2818 


2927 


2981 


54 


800 


3090 
3633 


3 144 


3199 


3253 


33o7 


336i 


3416 


3470 


3524 


35 7 8 


54 


801 


368 7 


374i 


37 9 5 
433 7 


3849 


3904 


3 9 58 


4012 


4066 


4120 


54 


802 


4i74 


4229 


4283 


4391 
4932 


4445 


4499 


4553 


4607 


4661 


54 


8o3 


4716 


4770 


4824 


4878 


4986 


5o4o 


5094 


5i48 


5202 


54 


804 


5256 


53io 


5364 


5418 


5472 


5D26 


558o 


5634 


5688 


5742 


54 


8o5 


5 7 o6 
6335 


585o 


5go4 


5 9 58 


6012 


6066 


6119 


6i 7 3 


6227 


6281 


54 


806 


638 9 


6443 


6497 


655i 


6604 


6658 


6712 


6766 


6820 


54 


807 
808 


6874 


6927 


6981 


7o35 


7089 


7i43 


7196 
7734 
8270 


725o 


73o4 


7358 


54 


74i 1 


7465 


7?'9 


7 5 7 3 


7626 


7680 
8217 


"87 


7841 


7895 
843i 


54 


809 


7949 


8002 


8o56 


8110 


8i63 


8324 


8378 


54 


810 


8485 


853 9 


85 9 2 


8646 


$3 


8 7 53 


8807 


8860 


8914 


8967 


54 


811 


9021 


9074 


9128 


9181 


9289 
9823 


9342 


9396 
9930 


9449 


9D03 


54 


812 


» 9336 


9610 


9663 


9716 


9770 


9877 


9984 
o5i8 


♦037 


53 


8i3 


91 0091 


0144 


0197 
073 1 


025l 


o3o4 


o358 


0411 


0464 


0571 


53 


814 


0624 


0678 


0784 


o838 


0891 


0944 


0998 


io5i 


1 104 


53 


8i5 


n58 


1211 


1264 


1317 


1371 


1424 


U77 


i53o 


1 584 


1637 


53 


816 


1090 


1743 


1797 


i85o 


1903 


ig56 


2009 


2o63 


2116 


2169 


53 


817 


2222 


2275 


2328 


238i 


2435 


2488 


2541 


2594 


2647 
3178 


2700 


53 


818 


2 7 53 


2806 


2859 


2913 


2966 


3019 


3072 


3i25 


323i 


53 


819 


3284 


333 7 


3390 


3443 
3 


3496 


3549 


36o2 


3655 


3708 


3761 


53 


N. 





1 


2 


4 


5 


6 


7 


8 


9 



14 




LOGARITHMS OF NUMBERS. Tabus L 


N. 





1 


2 


3 


4 


5 


6 


n 


8 


9 


D. 


820 


91 38i4 


386 7 


3920 


3973 


4026 


4079 

4608 


4i32 


4184 


423 7 


4290 


53 


821 


4343 


4396 


4449 


45o2 


4555 


4660 


4713 


4766 


4819 


53 


822 


4872 


4925 


4977 


5o3o 


5o83 


5i36 


5i8g 


5241 


52 9 4 


5347 


53 


823 


5400 


5453 


55o5 


5558 


061 1 


5664 


5716 


5769 


5822 


58 7 5 


53 


824 


5927 


5980 


6o33 


6o85 


6i38 


6191 


6243 


6296 


6349 


6401 


53 


825 


6454 


65o7 


655 9 


6612 


6664 


6717 


6770 


6822 


68 7 5 


6927 


53 


826 


6980 


7o33 


7085 


7 i38 


7190 


7243 


7295 


7348 


74oo 


7453 


53 


827 

828 


75o6 
8o3o 


7558 


761 1 


7663 


7716 


7768 


7820 


7873 


7925 


7978 


52 


8o83 


8i35 


8188 


8240 


8293 


8345 


8397 


845o 


8D02 


52 


829 


8555 


8607 


865g 


8712 


8764 


8816 


8869 


8921 


8 97 3 


9026 


52 


83o 


9078 


9i3o 


9 i83 


9235 


9287 


9340 


9 3 9 2 


9444 


9496 


9549 


52 


83 1 


*96oi 


9653 


9706 


9758 


9810 


9862 


9914 


9967 


♦019 


0071 


52 


832 


92 0123 


0176 


0228 


0280 


o332 


o384 


0436 


0489 


o54i 


0593 


52 


833 


o645 


0697 
1218 


0749 


0801 


o853 


0906 


0958 


1010 


1062 


1114 


52 


834 


1166 


1270 


l322 


i374 


1426 


1478 


i53o 


i582 


1 634 


52 


835 


1686 


1738 


1790 


1842 


1894 


1946 


1998 


2o5o 


2102 


2i54 


52 


836 


2206 


2258 


23lO 


2362 


24i4 


2466 


2Dl8 


2570 


2622 


2674 


52 


837 


2725 


2777 


2829 


2881 


2 9 33 


2985 


3o37 


3089 


3i4o 


3192 


52 


838 


3244 


3296 


3348 


3399 


345 1 


35o3 


3555 


3607 


3658 


3710 


52 


83 9 


3762 


38i4 


3865 


3917 


3969 


4021 


4072 


4124 


4176 


4228 


52 


840 


4279 


433i 


4383 


4434 


4486 


4538 


458 9 


4641 


4693 


4744 


52 


841 


479 6 


4848 


4899 


4951 


5oo3 


5o54 


5io6 


5i57 1 5209 


5201 


52 


842 


53i2 


5364 


54i3 


5467 


55i8 


5570 


5621 


5673 5720 
6188 1 6240 


5776 


52 


843 


5828 


5879 


5931 


5982 


6o34 


6o85 


6137 


6291 


5i 


844 


6342 


6394 


6445 


6497 


6548 


6600 


665i 


6702 


6754 


68o5 


5i 


845 


685 7 


6908 


6 9 5 9 
7473 


701 1 


7062 


7ii4 


7i65 


7216 


7268 


73i9 


5i 


846 


7370 


7422 


7324 

8037 


7 5 7 6 
8088 


7627 


7678 


773o 


778i 


7 832 


5i 


847 


7883 


7 9 35 


7986 


8140 


8191 


8242 8293 


8345 


5i 


848 


83 9 6 


8447 


8498 


8549 


8601 


8652 


8 7 o3 


8 7 54 


88o5 


8857 


5i 


849 


8908 


8 9 5 9 


9010 


9061 


9112 


9i63 


921D 


9266 


9 3i 7 


9368 


5i 


85o 


9419 


9470 


9521 


9 5 7 2 


9^23 


9674 


9725 


9776 


9827 


9879 


5i 


85i 


* 9930 


9981 


♦o32 


oo83 


oi34 


oi85 


0236 


0287 


o338 


o38 9 


5i 


852 


93 0440 


0491 


o542 


o5g2 


0643 


0694 


0745 


0796 


0847 


0898 


5i 


853 


0949 


1000 


io5i 


1102 


11 53 


1204 


1254 


i3o5 


i356 


1407 


5i 


854 


1458 


1 509 


i56o 


1610 


1661 


1712 


1763 


1814 


i860 


1915 


5i 


855 


1966 


2017 


2068 


2118 


2169 


2220 


2271 


2322 


2372 


2423 


5i 


856 


2474 


2524 


2575 
3o82 


2626 


2677 


2727 


2778 


2829 


2879 


2930 


§1 


857 


2981 


3o3i 


3i33 


3i83 


3234 


3285 


333d 


3386 


3437 


5i 


858 


3487 


3538 


3589 


363o 


3690 


3740 


3791 


3841 


38 9 2 


3 9 43 


5i 


85 9 


3993 


4044 


4094 


4145 


4195 


4246 


4296 


4347 


4397 


4448 


5i 


860 


4498 


4549 


4599 


465o 


4700 


475i 


4801 


4852 


4902 


4953 


5o 


861 


5oo3 


5o54 


5io4 


5i54 


52o5 


5255 


53o6 


5356 


5406 


5457 


5o 


862 


55o7 


5558 


56o8 


5658 


5709 


5759 


5809 
63 13 


5S6o 


5910 


5960 


5o 


863 


601 1 


6061 


6111 


6162 


6212 


6262 


6363 


64 1 3 


6463 


5o 


864 


65i4 


6564 


6614 


6665 


6713 


6 7 65 


68i5 


6865 


6916 


6966 


5o 


865 


7016 


7066 


7117 


7161 

7668 


7217 


7267 


7317 


736 7 


74i8 


7468 


5o 


866 


7D18 
8019 


7568 


7618 


7718 


7769 
8269 


7819 
8320 


7869 
8370 


7919 


7969 


5o 


867 
868 


8069 


8119 


8169 


8219 


8420 


8470 


5o 


8520 


8570 


8620 


8670 


8720 


8770 


8820 


8870 8920 


8970 


5o 


86 9 


9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 j 9419 


9469 


5o 


870 


9 5io 
940018 


q569 


9610 
0118 


9669 
0168 


9719 
0218 


9769 


9819 


9S69 '9918 


9968 


5o 


871 


0068 


0267 


o3i7 0367 0417 


0467 


5o 


872 


o5i6 


o566 


0616 


0666 


0716 


0765 


081 5 ; oS65 0915 


0964 


5o 


8 7 3 


1014 


1064 


1 1 14 


n63 


I2l3 


1263 


i3i3 


i362 . 1412 


1462 


5o 


874 


i5ii 


i56i 


1611 


1660 


1710 


1760 


1809 


1809 1909 


1958 


5o 


875 


2008 


2o58 


2107 


2157 


2207 


2256 


23o6 


2355 24o5 


2455 


5o 


876 


25o4 


2554 


26o3 


2653 


2702 


2752 


2801 1 2S5i 2901 


2950 


5o 


877 


3ooo 


3 049 3099 


3i48 


3198 


3247 


3297 3346 3396 


3445 


49 


878 


3495 
3 9 8 9 


3544 


35 9 3 


3643 


36o2 


3742 


3791 j 384i 3Soo 


3939 
4433 


49 


87Q 


4o38 


4088 


4i3 7 


4106 


4236 


4285 ; 4335 43b4 


49 


N. 





1 


2 


3 


4 


■ 


6 | n 1 8 j 9 


D. 



Table I. 


LOGARITHMS OF NUMBERS. 15 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


880 


94 4483 


4532 


458i 


463 1 


4680 


4729 


4779 


4828 


4877 


4927 


49 


881 


4976 


5o25 


5074 


5i24 


5i 7 3 


5222 


5272 


532i 


5370 


5419 


49 


882 


5469 


55i8 


556 7 


56i6 


5665 


5 7 i5 


5764 


58i3 


5862 


5912 


49 


883 


5 9 6i 


6010 


6059 


6108 


6157 


6207 

6698 


6256 


63o5 


6354 


64o3 


49 


884 


6402 


65oi 


655i 


6600 


6649 


6747 


6796 


6845 


6894 


49 


885 


6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7336 


7385 


49 


886 


7434 


7483 


7532 


758! 


763o 


til 


7728 


7777 
8266 


7826 


7875 


49 


887 


l 9 H 
84i3 


7973 


8022 


8070 


8119 


8217 


83i5 


8364 


49 


888 


8462 


85n 


856o 


8609 


8657 


8706 


8 7 55 


8804 


8853 


49 


889 


8902 


8 9 5 1 


8999 


9048 


9097 


9146 


9 i 9 5 


9244 


9292 


9341 


49 


Poo 


9390 


9439 


9488 


9 536 


9585 


9634 


9 683 


973i 


9780 


9829 


49 


891 


^9878 


9926 


9975 


♦024 


0073 


0121 


0170 


0219 


0267 


o3i6 


49 


892 


95o365 


0414 


0462 


o5u 


o56o 


0608 


0657 


0706 


0754 


o8o3 


49 


8 9 3 


o85i 


0900 


0949 
1435 


ffl 


1046 


1095 
i58o 


1 U3 


1 192 


1240 


1289 
1775 


49 


894 


1338 


1 386 


i532 


1629 


1677 


1726 


49 


890 


1823 


1872 


1920 


;$ 


2017 


2066 


2114 


2i63 


2211 


2260 


48 


896 


23o8 


2356 


24o5 


2502 


255o 


3oS 


2647 


2696 
3i8o 


2744 


48 


897 


2792 


2841 


2889 
33 7 3 


2 9 38 


2986 


3o34 


3i3i 


3228 


48 


898 


3276 


3325 


3421 


3470 


35i8 


3566 


36i5 


3663 


37 1 1 


48 


899 


3760 


38o8 


3856 


3905 


3953 


4001 


4049 


4098 


4146 


4194 


48 


900 


4243 


4291 


4339 


4387 


4435 


4484 


4532 


458o 


4628 


4677 


48 


901 


4725 


4773 


4821 


4869 


4918 


4966 


5oi4 


5o62 


5no 


5i58 


48 


902 


6207 


5255 


53o3 


535i 


5399 


5447 
5928 


5495 


5543 


5592 


564o 


48 


903 


5688 


5736 


5784 


5832 


588o 


5976 


6024 


6072 


6120 


48 


904 


6168 


6216 


6265 


63i3 


636 1 


6409 


6457 


65o5 


6553 


6601 


48 


905 


66^9 
7128 


6697 


6745 


6793 


6840 


6888 


6 9 36 


6984 


703 2 


7080 


48 


906 


7176 
7 655 


7224 


7272 


7320 


7368 


74i6 


7464 


7512 


7 55o 
8o38 


48 


907 


8086 


7 7 o3 


775i 


7799 


7847 


7894 


7942 


7990 
8468 


48 


908 


8i34 


8181 


8229 


8277 


8325 


83 7 3 


8421 


85i6 


48 


909 


8564 


8612 


865 9 


8707 


8755 


88o3 


885o 


8898 


8946 


8994 


48 


910 


9041 


9089 


9 i3 7 


9i85 


9232 


9280 


9328 


9375 


9423 


9471 


48 


911 


9 5i8 


9 566 


9614 


9661 


9709 
oi85 


9757 


9804 


9852 


9900 


9947 


48 


912 


* 9995 


♦042 


0090 


oi38 


0233 


0280 


o328 


0376 


0423 


48 


9i3 


960471 


o5i8 


o566 


o6i3 


0661 


0709 


0756 


0804 


o85i 


0899 


48 


914 


0946 


0994 


1041 


1089 


n36 


1 184 


I23l 


1279 


i326 


1374 


47 


9 i5 


1421 


1469 
1943 


i5i6 


1 563 


1611 


1 658 


1706 


i 7 53 


1801 


1848 


47 


916 


i8 9 5 


1990 


2o38 


2085 


2l32 


2180 


2227 


2275 


2322 


47 


917 


2369 

2843 


2417 


2464 


25ll 


2559 


2606 


2653 


2701 


2748 


27 9 5 


47 


918 


2890 


2 9 3 7 


2985 


3o32 


3o 79 

3552 


3 1 26 


3i74 


3221 


3268 


47 


919 


33i6 


3363 


34io 


3457 


35o4 


3599 


3646 


36 9 3 


3741 


47 


920 


3788 


3835 


388a 


3929 


3977 


4024 


4071 


41 18 


4i65 


4212 


47 


921 


4260 


4307 


4354 


4401 


4448 


4495 


4542 


4590 


463 7 


4684 


47 


922 


473i 


4778 


4825 


4872 


4919 
5390 


4966 


5oi3 


5o6i 


5io8 


5i55 


47 


923 


5202 


5249 


5296 


5343 


5437 


5484 


553 1 


55 7 8 


5625 


47 


924 


5672 


5719 


5766 


58i3 


586o 


5907 


5954 


6001 


6048 


6095 


47 


925 


6l42 


6189 


6236 


6283 


6329 


63 7 6 


6423 


6470 


65i 7 


6564 


47 


926 


66l I 


6658 


6705 


6 7 5z 


6799 


6845 


6892 


6 9 3o 


6986 


7o33 


47 


927 


7080 


7127 


7173 


7220 


7267 


73i4 


736i 


7408 


7454 


75oi 


47 


928 


7548 


75 9 5 
8062 


7642 


7688 
8i56 


7735 


7782 


7829 


7875 


7922 
83 9 o 


7969 


47 


929 


8016 


8109 


82o3 


8249 


8296 


8343 


8436 


47 


93o 


8483 


853o 


85 7 6 


8623 


8670 


8716 


8 7 63 


8810 


8856 


8903 
9369 


47 


93 1 


8 9 5o 


8996 


9043 


9090 


9i36 


9i83 


9229 


9276 


9 323 


47 


932 


94i6 


9463 


9509 


9 556 


9602 


9649 


9695 


9742 


9789 


9 835 


47 


9 33 


* 9 882 


9928 


9973 


♦021 


0068 


0114 


0161 


0207 


0254 


o3oo 


% 


934 


97 °347 


0393 


0440 


0486 


o533 


o5 79 


0626 


0672 


0719 


0765 


9 35 


0812 


o858 


0904 
1369 


0951 


0997 


1044 


1090 
1 554 


u3 7 


u83 


1229 
1693 


46 


9 36 


1276 


l322 


Ui5 


1461 


i5o8 


1601 


1647 


46 


9 3 7 
9 38 


1740 


1786 


i832 


1879 


1925 

2388 


1971 


2018 


2064 


2110 


2107 


46 


2203 


2249 


2295 


2342 


2434 


2481 


2527 


25 7 3 


2619 


46 


9 3 9 


2666 


2712 


2 7 58 


2804 


285i 


2897 


2943 


2989 


3o35 


3082 


46 


N. 





1 


2 


» 1 4 


6 


6 


7 


8 


9 


D. 



16 




LOGARITHMS OF NUMBERS. Table L 


X. 





1 | 2 | 3 | 


4 


5 


6 | 7 | 8 


9 


D. 


94o 


97 3i28 


3i74 


3220 


3266 


33i3 


3359 


34o5 345 1 i 3497 


3543 


46 


941 


3390 


3636 


3682 


3728 


3774 


3820 


3866 j 3913 1 3959 
4327 4374 | 4420 


400 5 


46 


942 


4031 


4097 


4i43 


4189 


4235 


4281 


4466 


46 


943 


4312 


4558 


4604 


465o 


4696 


4742 


4788 J 4834 ! 4880 


4926 


46 


944 


4972 


5oi8 


5o64 


5uo 


5i56 


5202 


5248 | 5294 j 534o 


5386 


46 


945 


5432 


5478 


5524 


5570 


56i6 


5662 


3707 5753 ! 5799 


5845 


46 


946 


58 9 i 


5 9 3 7 
63 9 6 


5 9 83 


6029 


6073 


6121 


6167 6212 , 6238 


63o4 


46 


947 


635o 


6442 


6488 


6533 


6579 


6625 6671 j 6717 


6763 


46 


948 


6808 


6854 6900 


6946 


6992 


7o3 7 


7083 


7129I 7175 


7220 


46 


949 


7266 


73i2 


7358 


74o3 


7449 


7495 


7541 


7386 j 7632 


7678 


46 


930 


7724 


7769 
8226 


7 8i5 


7861 


7906 


79 52 


7998 


8043 ! 8089 


8i35 


46 


931 


8181 


8272 


8317 


8363 


8409 


8454 


85oo 8546 


85gi 


46 


932 


8637 


8683 


8728 


8774 


8819 


8865 


891 1 


8956 9002 


9047 


46 


953 


9093 


9 i38 


9184 


923o 


9275 


9321 


9 366 


9412 9457 


95o3 


46 


9^4 


9348 


9394 


9639 


9 685 


97 3 ° 


9776 


9821 


9867 


9912 


99 58 


46 


955 


98 ooo3 


0049 


0094 


0140 


oi85 


023l 


0276 


0322 


0367 


0412 


45 


936 


0458 


o5o3 


0549 
ioo3 


0594 


0640 


0685 


0730 


0776 


0821 


0867 


45 


937 


0912 


0937 


1048 


1093 


1 i3g 


1 184 


1229 


1275 


1 3 20 


45 


9 58 


1 366 


1411 


1456 


i5oi 


1 547 


1592 


1637 


1683 1728 


1773 


45 


9 5 9 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2i35 2181 


2226 


45 


960 


2271 


23i6 


2362 


2407 


2452 


2497 


2543 


2588 ! 2633 


2678 


45 


961 


2~23 


2769 


2814 


2859 


2904 

3356 


2949 


2994 3o4o 3o85 


3i3o 


45 


962 


3l75 


3220 


3265 


33io 


3401 


3446 


3491 


3536 


358i 


45 


9 63 


3626 


36 7 i 


3 7 i6 


3762 


3807 


3852 


3^97 


3 9 42 
4392 


3987 


4o32 


45 


964 


4077 


4122 


4167 


4212 


4257 


43o2 


4347 


4437 


4-4=2 


45 


963 


4527 


4572 


4617 


4662 


4707 


4732 


4797 


4842 


4887 


4932 
5382 


43 


966 


4977 


5022 


5067 


5lI2 


5i57 


5202 


5247 


5292 


5337 


45 


967 


5426 


5_47I 


55i6 


556 1 


56o6 


565 1 


5696 


5741 


5 7 86 


583o 


45 


968 


5875 


5920 


5965 


6010 


6o55 


6100 


6i44 


6189 


6234 


6279 


45 


969 


6324 


636 9 


641 3 


6458 


65o3 


6548 


6593 


6637 


668;, 


6727 


45 


9"o 


6772 


6817 


6861 


6906 
7353 


6951 
7393 


6996 


7040 


7085 7130 


7175 


45 


971 


7219 


7264 


7309 


7443 


7488 


7532 | 7077 


7622 


45 


972 


7666 


7711 


7756 
8202 


7800 


7845 


789c 


79 34 


-9-9 8024 


8068 


45 


973 


81 13 


807 


8247 


8291 


8336 


838i 


8423 1 8470 


85i4 


45 


974 


8539 


8604 


8648 


8693 


8 7 3 7 


8782 


8826 


8871 8916 


8960 


43 


973 


9005 


9049 


9004 


9»38 


9183 


9227 


9272 


9316] 9361 


94o5 


45 


9" 6 


9430 


9494 


o53g 
9 9 83 


9583 


9628 


9672 


9-17 


9761 1 9S06 


985o 


44 


977 


* 9S95 

99 0339 

0783 


9939 


♦028 


0072 


0117 


0161 


0206 j 0250 


0294 

073S 


44 


978 


o3S3 


0428 


0472 


o5i6 


o56i 


o6o5 


0030 0694 


44 


979 


0S27 


0871 


0916 


0960 


1004 


1049 


1093 1137 


11S2 


44 


980 


1226 


1270 


i3i5 


i359 


i4o3 


144S 


U92 
1935 

23 77 


1 536 


i58o 


i625 


44 


981 


1669 


1713 


1758 


1802 


1846 


ibQO 


1979 


2023 


2067 


44 


982 


21 11 


2 1 56 


2200 


2244 


2288 


2333 


2421 


2465 


2 509 


44 


9 83 


2334 


2398 


2642 


2686 


2730 


2774 


2S19 


2863 


2007 
3348 


2951 


44 


984 


2 99 5 


3o3g 


3o83 


3127 


3172 


32i6 


3200 


33o4 


3392 


44 


9 85 


3436 


348o 


3524 


3568 


36i3 


365 7 


3701 


3 7 45 


3789 


3833 


44 


986 


38 77 


3921 
436 1 


3965 


4009 


4o53 


4097 
453 7 


4141 


4i85 4229 


4273 


44 


987 


43i7 


44o5 


4449 


44g3 


458i 


4623 4669 
5o65 5io8 


47i3 


44 


988 


4757 


4801 


4845 


48S9 
5328 


4933 


4977 


5021 


5i52 


44 


989 


5ig6 


5240 


5284 


5372 


5416 


546o 


55o4 i 5547 


5591 


44 


990 


5635 


56 79 


5 7 23 


5767 


58n 


5854 


58o8 


5942 ' 59S6 


6o3o 


44 


991 


6074 


6117 


6161 


62o5 


6249 


6293 


6337 63So 6424 


44 


992 


65i2 


6555 


65 99 


6643 


6687 


6731 


6774 6S1S 6862 6906 


-i 


9 9 3 


6949 
7386 


6993 


7037 1 7080 


7124 


7168 


7212 | 7255 ! 7299 7343 

764S 7C92 7700 7779 


44 


994 


743o 


7474 7317 


756i 


7605 


44 


993 


7823 


•7S67 
83o3 


7910 7954 
8347 83 Q o 


ss 


804 1 


8oS5 8129 S1-2 8216 


44 


996 


8239 


8477 


S52i 8304 S60S B655 


44 


997 


8693 


8739 ! 8782 8b26 


8869 


8913 
9 348 


S956 9000 9043 9087 


44 


998 


9i3i 


9174 9218 9261 


93o3 


9092 9435 94~9 9322 


44 


999 


y565 


9609 9652 9696 


9739 


9783 


9S26 9S70 9913 9937 


.3 


N. 





1 i 2 | 3 


4 


5 


6 7 i S | 9 


D. 

1 



TABLE II 



LOGARITHMIC SINES AND TANGENTS 



EVERY DEGREE AND MINUTE OF THE aUADRANT. 



If tne logarithms of the values in Table III. be each increased by 10, the results 
w ill be the values of this table. 

The logarithmic Secants and Cosecants are not given. They may be readily ob- 
tained, as follows : — Subtract the logarithmic Cosine from 20, and the remainder 
will be the logarithmic Secant ; subtract the logarithmic Sine, from ?o, and the 
remainder will be the logarithmic Cosecant. 



38 



18 


LOGARITHMIC SINES, 


TANGENTS, ETC 


Table 


II. 


0° 














179° 


i 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 





Inf. Neg. 




1 • 000000 




Inf. Neg. 




Infinite. 


60 


i 


6.463726 


5oi 717 


000000 


00 


6.463726 


501717 


13.536274 


M 


2 


764756 


293485 


000000 


00 


764756 


293483 


235244 


3 


940847 


208231 


000000 


00 


940847 


2o823l 


059 1 53 


57 


4 


7-065786 


161D17 


000000 


00 


7.065786 


161517 


12-934214 


56 


5 


162696 


l3io68 
111675 


000000 


00 


162696 


131969 
111378 


8373o4 


55 


6 


241877 


9.999999 


01 


241878 


758122 


54 


7 


3o8824 


o6653 


999999 


01 


3o8825 


99653 
85254 


691173 


53 


8 


3668i6 


85254 


999999 


01 


3668l 7 


633i83 


52 


9 


417968 


76263 


999999 
999998 


01 


417970 


76263 


582o3o 


5i 


10 


463726 


68988 


01 


463727 


68988 


536273 


5o 


ii 


7«5o5n8 


62981 


9.999998 


01 


7»5o5i2o 


62981 


12-494880 


% 


12 


542906 


5 79 36 


999997 


01 


542909 


57933 


457091 


i3 


577668 


53641 


990997 
999996 


01 


577672 


53642 


422328 


47 


i4 


609853 


49938 


0: 


609857 


49939 


390143 


46 


i5 


639816 


46714 


999996 


01 


639820 


46713 


36oi8o 


45 


16 


667845 


4388i 


Q9999 5 


01 


667849 


43882 


332i5i 


44 


17 


694173 


41372 


999993 


01 


694179 
719003 


4i373 


3o582i 


43 


18 


718997 


3gi35 


999994 


01 


39i36 


280997 


42 


»9 


742478 


37127 


99999 3 


01 


742484 


37128 


257316 


4i 


20 


764734 


353i5 


999993 


01 


764761 


35i36 


235239 


40 


21 


7-785943 


33672 


9.999992 


01 


7.785951 


33673 


12 .214049 


39 


22 


806146 


32175 


999991 


01 


8061 55 


32176 


193843 


38 


23 


82545i 


3o8o5 


999990 
999989 


01 


825460 


3o8o6 


H4540 


37 


24 


843934 


29547 
28388 


02 


843944 


29549 


i56o56 


36 


25 


861662 


999989 
999988 


02 


861674 


28390 


138326 


35 


26 


878695 


27317 


02 


878708 


27318 


1 21 292 


34 


27 


895083 


26323 


999987 


02 


895099 


26325 


1 0490 1 


33 


28 


910879 


25399 
24538 


999986 


02 


910894 
926134 


254oi 


089106 


32 


29 


9261 19 


999983 


02 


24540 


073866 


3i 


3o 


940842 


23733 


999983 


02 


94o858 


23735 


039142 


3o 


3i 


7.955082 


22980 


9.999982 


02 


7.955100 


22981 


1 2 • 044900 


3 


32 


068870 


22273 


99998i 


02 


968889 
982253 


22275 


o3i 1 1 1 


33 


982233 


21608 


999980 


02 


21610 


017747 


27 


34 


995198 
8-007787 


20981 
20390 
1 9 83 1 


999979 


02 


995219 


2oo83 
20392 
l 9 833 


004781 


26 


35 


999977 


02 


8.007809 


11-992191 


*5 


36 


020021 


999976 


02 


020044 


970936 


24 


37 


031919 


19302 


999973 


02 


031945 


l93o5 
i8So3 


968o55 


23 


38 


043 DO I 


18801 


999973 


02 


043527 


936473 


22 


3 9 


05478l 


18325 


999972 


02 


054809 


1 S3 27 


943191 


21 


4c 


065776 


17872 


999971 


02 


o658o6 


17874 


934194 


20 


4i 


8-076500 


17441 


9.999969 
999968 


02 


8-076531 

086997 


17444 


11-923460 
9i3oo3 


IQ 


42 


086963 


1 703 1 


02 


17034 


18 


43 


O97183 


16639 


999966 


02 


097217 


16642 


902783 
802797 
8S3o37 


»7 


44 


I07l6"7 
I 16926 


16263 


999964 


03 


107203 


16268 


16 


45 


15908 


999963 


o3 


1 16963 


15910 


i5 


46 


126471 


15566 


99996i 


o3 


I263I0 


1 5568 


873490 


14 


% 


i358io 


15238 


999939 
999958 


o3 


i3585i 


i524i 


864149 


i3 


1M953 


14924 


o3 


144996 


14927 


855oo4 


12 


49 


153907 


14622 


999956 


o3 


i53y32 


14627 


846048 


11 


5o 


162681 


14333 


999934 


o3 


162727 


14336 


837273 


I0 « 


5i 


8-171280 


Uo54 


9- 999952 


o3 


8-171328 


i4o57 


11-828672 


\ 


52 


I797i3 


i3 7 86 


9999 3 ° 


o3 


i88o36 


13790 


820237 


53 


187985 


i3529 


999948 


o3 


13532 


81 1964 
8o3844 


1 


54 


196102 


i328o 


999946 


o3 


196106 


13284 


6 


55 


204070 


l3o4i 


999944 


o3 


204126 


i3o44 


795874 


5 


56 


2ii8o5 


12810 


999942 


04 


21 1953 


12814 


788047 


4 


57 
58 


219581 
227134 


12587 
12372 


999940 
999938 


04 
04 


219641 
227195 


12390 
12376 


7S0359 
772800 


3 
2 


5g 


234557 
24i855 


12164 


999936 


04 


234621 


12168 


765379 


1 


6o 


1 1963 


999934 


04 


241921 


11967 


758079 





t 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


90< 


) 














89° 



Table IL 


LOGARITHMIC SINES, TANGENTS, ETC. 19 


1° 














178° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


t 





8-24i855 


iiq63 


9.999934 


04 


8-241921 


1 1967 


11.758079 
750898 
743835 


60 


i 


249033 


11768 


999 9 32 


04 


249102 


11772 


% 


2 


256094 


n58o 


999929 


04 


256i65 


1 1 584 


3 


263o42 


1 1 398 


999927 
999925 


04 


263u5 


1 1402 


736885 


n 


4 


269881 


11221 


04 


269956 


1 1225 


73oo44 


5 


276614 


no5o 


999922 


04 


276691 


uo54 


723309 


55 


6 


283243 


io883 


999920 


04 


283323 


10887 


716677 


54 


7 


289773 


10721 


999918 


04 


289856 


10726 


710144 


53 


8 


296207 


io565 


999915 


04 


296292 
3o2634 


10570 


703708 


52 


9 


3o2D4o 


io4i3 


999913 


04 


10418 


697366 


5i 


10 


308794 


10266 


9999IO 


04 


3o8884 


10270 


691116 


5o 


ii 


8.314954 


10122 


9.999907 


04 


8-3i5o46 


10126 


11.684954 

678878 
672886 


% 


12 


321027 
327016 


9982 


999905 


04 


321 122 


9087 


i3 


9847 


999902 


04 


327114 
333o25 


9801 


47 


U 


332924 


9714 


999899 


o5 


9719 


666975 


46 


i5 


338 7 53 


9 586 


999897 


o5 


338856 


9590 


661 144 


45 


16 


3445o4 


9460 


999894 


o5 


344610 


9465 


655390 


44 


\l 


35oi8i 


9 338 


999891 


o5 


350289 
3558q5 
36i43o 


9 343 


649711 


43 


355783 


9210 
91 o3 
8990 


999888 


o5 


9224 


644105 


42 


*9 


36i3i5 


999885 


o5 


9108 
8 99 5 


6385 7 o 


4i 


20 


366777 


999882 


o5 


3668 9 5 


633io5 


40 


21 


8-372171 


8880 


9.999879 


o5 


8-372292 


8885 


11.627708 


IS 


22 


377499 


8772 


999876 


o5 


377622 
382889 


8777 


622378 


23 


382762 


8667 


999873 


o5 


8672 


617111 


37 


24 


387962 


8564 


999870 


o5 


388092 
393234 


8570 


61 1908 


36 


25 


3g3ioi 


8464 


999867 


o5 


8470 


606766 


35 


26 


398179 


8366 


999864 


o5 


3983i5 


83 7 i 


6oi685 


34 


3 


403199 


8271 


999861 


o5 


4o3338 


8276 


596662 


33 


408161 


8177 


999858 


o5 


4o83o4 


8182 


591696 
586787 


32 


29 


4i3o68 


8086 


999854 


o5 


4i32i3 


8091 


3i 


3o 


4i79 J 9 


7996 


999851 


06 


418068 


8002 


58ig32 


3o 


3i 


8-422717 


7909 


9.999848 


06 


8-422869 
427618 


7914 
783o 


n-577i3i 


3 


32 


427462 


7823 


999844 


06 


572382 


33 


432 1 56 


7740 


999841 


06 


4323 1 5 


7745 


567685 


27 


34 


4368oo 


7 65 7 


999838 


06 


436962 


7663 


563o38 


26 


35 


44i394 


7577 


999834 


06 


44i56o 


7583 


558440 


25 


36 


445941 


7499 


99983 1 


06 


4461 10 


75o5 


5538 9 o 
549387 


24 


3? 


45o44o 


7422 


999827 


06 


45o6i3 


7428 


23 


38 


454893 


7346 


999824 


06 


455070 


7 35 2 


544930 
540619 


22 


39 


459301 

463665 


7273 


999820 


06 


459481 
46J849 


7279 


21 


4o 


7200 


999816 


06 


7206 


536i5i 


20 


4i 


8-467985 


7129 


9-999813 


06 


8-468172 


7i35 


u-53i828 


\l 


42 


472263 


7060 


999809 
999805 


06 


472454 


7066 


527546 


43 


476498 


6991 


06 


476693 


6998 
6931 


5233o7 
519108 


17 


44 


480693 


6924 


999801 


06 


480892 


16 


45 


484848 


685 9 


999797 


07 


485oDo 


6865 


5 1 4950 


i5 


4b 


488963 


6794 


999794 


07 


489170 


6801 


5io83o 


14 


% 


493o4o 


6 7 3i 


999790 
999786 


07 


49325o 


6738 


5o675o 


i3 


497078 


6669 
6608 


07 


497293 


6676 


502707 


12 


49 


5oio8o 


999782 


07 


501998 


66i5 


498702 


11 


5o 


5o5o45 


6548 


999778 


07 


50^267 


6555 


494733 


10 


5i 


8.508974 


6489 


9*999774 


07 


8-5oo2oo 
51J098 
516961 


6496 


11-490800 


I 


52 


512867 
516726 


643 1 


999769 


07 


6439 


486902 


53 


63 7 5 


999765 


07 


6382 


483o39 


7 


54 


52o55i 


63 19 


999761 


07 


520790 
524586 


6326 


479210 
475414 


6 


55 


524343 


6264 


999757 
999753 


07 


6272 


5 


56 


528102 


621 1 


07 


528349 


6218 


47i65i 


4 


57 


53i828 


6i58 


999748 


07 


532o8o 


6i65 


467920 


3 


58 


535523 


6106 


999744 


07 


535779 


6u3 


464221 


2 


5 9 


53gi86 


6o55 


999740 


07 


539447 
543o84 


6062 


46o553 


1 


60 


542819 


6004 


999735 


07 


6012 


456916 




t 


/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


91 c 














88° 



20 


LOGARITHMIC SINES, 


TANGENTS, ETC 


Tabi e 


1L 


2 3 














1 


77° 


/ 


Sine. 


D - 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 

60 





8042819 


G004 


9-999735 


07 


8- 543o84 


6012 


II-4569I6 
453309 


i 


54642 2 


5955 


999731 


07 


546691 


5962 


i 


2 


549995 
553539 


5 9 o6 


999726 


3 


550268 


5914 


449732 


3 


5858 


999722 


553817 


5866 


446 1 83 


57 


4 


557054 


58u 


999717 
999713 


08 


557336 


5819 


442664 


56 


5 


56o54o 


5 7 65 


08 


56o828 


5773 


439172 


55 


6 


563999 
56 7 43 1 


5 7 i 9 


999708 


o3 


564291 


5727 


435709 


54 


I 


56 7 4 


999704 


08 


567727 


5682 


432273 
428863 


53 


570836 


563o 


999699 


08 


57Il37 


5638 


52 


9 


574214 


5587 


999604 
999689 


08 


574520 


5595 


425480 


5i 


10 


577566 


5544 


08 


577877 


5532 


422123 


5o 


n 


8.58oS 9 2 


55o2 


9-999680 


o3 


8081208 


55io 


II-4I8792 
4l54»6 


% 


12 


584193 


546o 


999680 


08 


5845 14 


5468 


i3 


587469 


5419 


999670 


08 


587795 


5427 


412205 


47 


U 


090721 


53-9 


999670 


08 


59io5i 


538 7 


408949 


46 


ID 


593948 


5339 


999660 


08 


5 9 4283 


5347 


405717 
4o25o8 


45 


16 


597132 


53oo 


999660 


08 


597492 


53o8 


44 


17 


6oo332 


526i 


999655 


08 


600677 


5270 


399323 


43 


18 


603489 


0223 


999650 


08 


6o3S39 
606978 


5232 


396161 


42 


19 


606623 


5i86 


999645 


09 


5ig4 


393022 
389906 


4i 


20 


609734 


5i49 


999640 


09 


610094 


5i58 


40 


21 


8-612823 


5lI2 


9-999635 


09 


8-613189 


5l21 


ii-3868ii 


ll 


22 


610891 
618937 


5076 


999629 


09 


616262 


5o85 


383 7 38 


23 


5o4i 


999624 


09 


619313 


5o5o 


380687 


37 


24 


621962 


5oo6 


999619 


09 


622343 


5oi5 


377657 
374648 


36 


25 


624960 


4972 


999614 


09 


625352 


49S1 


35 


26 


627948 


4933 


999608 


09 


628340 


4947 


371660 


34 


27 


630911 

633854 


4904 


999603 


09 


63i3o8 


4913 


3686 9 2 


33 


28 


4871 


999597 


09 


634256 


4880 


365744 


32 


29 


636776 


483 9 


999092 


09 


637184 


4848 


362816 


3i 


3o 


63 9 68o 


4806 


999086 


09 


640093 


4816 


359907 


5: 


3i 


8-642563 


4775 


9 -99 9 58i 


09 


8-642982 


4784 


11-357018 


2 


32 


645423 


4743 


999575 


09 


645853 


4753 


354147 
35i2o6 


33 


648274 


4712 


999370 


09 


648704 


4722 


27 


34 


65uo2 


4682 


999364 


09 


65i537 


4691 


348463 


26 


35 


65391 1 


4652 


99g558 


10 


654352 


4661 


345648 


25 


36 


656702 


4622 


999553 


10 


657149 
659928 


463 1 


34255i 


24 


37 


659470 


4592 


999547 


10 


4602 


340072 


23 


33 


66223o 


4563 


999541 


10 


662689 
665433 


4573 


33 7 3n 


22 


39 


664968 


4535 


999535 


10 


4544 


334567 


21 


4o 


667689 


45o6 


999529 


10 


66S160 


4526 


33i84o 


20 


4i 


8.6 7 o3 9 3 


44-9 


9-999324 


1: 


8-670870 


4488 


n-329i3o 


8 


42 


673obo 


445i 


9995 1 8 


10 


673563 


446i 


326437 


43 


675751 


4424 


999312 


10 


676239 


4434 


323761 


3 


44 


678405 


4397 


999506 


10 


678900 


4417 


32II00 


45 


68io43 


4370 


999500 


10 


68 1 544 


438o 


3 1 8456 


i5 


46 


683665 


4344 


999493 
9994^7 


10 


684172 
686784 


4354 


3i5828 


14 


47 


6S6272 


43i8 


1 : 


4328 


3i32i6 


i3 


48 


688863 


42Q2 


999481 


10 


6So38i 


43o3 


310619 


12 


49 


691438 


4267 


999475 




691963 


4277 


3o3o37 


11 


5o 


693998 


4242 


999469 


10 


694029 


4252 


3o547i 


10 


5i 


8.696543 


4217 


9-999463 


11 


8-697081 


4228 


11-302919 
3oo383 


i 


52 


6990-3 
701589 


4192 


999456 


11 


699617 


42o3 


53 


4168 


99945o 


11 


702139 
704646 


417^ 


297861 


7 


54 


704090 


4i44 


999443 


11 


4i55 


293334 


6 


55 


706577 


4121 


999437 


11 


707140 


4i32 


292860 


5 


56 


709049 


4097 


99943i 


11 


709618 


41 oS 


290382 


4 


57 


71 1007 


40-4 


999424 


11 


712083 


4oS5 




3 


58 


7i3g52 


4o5i 


999418 


11 


7U534 


4062 


2S5466 


2 


5 9 


716333 


4029 
4006 


99941 1 


11 


716972 


4040 


2S002S 


1 


00 


718803 


999404 


II 


719396 


4017 


2S0604 





' 


Cosiae. 


»• 


Sine. 


D. 


Cotang. 


D. 


Tang. 


1 


92° 
















\'° 



Table II. LOGARITHMIC SINES, TANGENTS, ETC. 21 


3° 














176° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 


o 


8-718800 


4006 


9-999404 


II 


3.719396 


4017 
3993 


1 1 - 280604 


60 


i 


721204 


3984 


999398 


11 


721806 


278194 


n 


2 


723595 


3962 


999391 


1 1 


724204 


3 97 4 


275796 


3 


725972 
728337 


3941 


999384 


u 


726588 


3 9 52 


273412 


n 


4 


l%l 


999378 


n 


728959 
73i3i7 


3930 


27 1 041 


5 


730688 


999371 


11 


3909 
388 9 
3868 


268683 


55 


6 


733027 


38 7 7 
385 7 


999364 


12 


733663 


266337 


54 


7 


735354 


999357 


12 


735996 


264004 


53 


8 


737667 


3836 


99935o 


12 


7383i7 
740626 


3848 


26i683 


52 


9 


739969 


38i6 


999343 


12 


3827 


259374 


5i 


10 


742259 


3796 


999336 


12 


742922 


3807 


257078 


5o 


ii 


8-744536 


3776 


9.999329 


12 


3.745207 


3787 
3768 


1 r • 254793 


% 


12 


746802 


3 7 56 


99g322 


12 


747479 


252521 


i3 


749055 


3 7 3 7 


9993 1 5 


12 


749740 


3749 


250260 


47 


14 


751297 
753528 


3717 


999308 


12 


751989 


3729 


24801 1 


46 


i5 


36 9 8 


999301 


12 


754227 


3710 


245 77 3 


45 


16 


755747 


3679 


999294 
999287 


12 


756453 


3692 


243547 


44 


17 


757955 


366i 


12 


758668 


3673 
3655 


24i332 


43 


18 


7601 5 1 


3642 


999279 


12 


760872 


239128 


42 


19 


762337 


3624 


999272 


12 


763o65 


3636 


236935 


4i 


20 


7645 1 1 


36o6 


999265 


12 


765246 


36i8 


234754 


40 


21 


8-766675 


3588 


9-999257 


12 


8-767417 


36oo 


11-232583 


is 


22 


768828 


3570 


999250 


i3 


769578 


3583 


230422 


23 


770970 


3553 


999242 


i3 


771727 


3565 


228273 
226134 


U 


24 


773101 


3535 


999235 


i3 


773866 


3548 


25 


775223 


35i8 


999227 


i3 


775 99 5 


353i 


224005 


35 


26 


777333 


35oi 


999220 


i3 


7781 i4 


35i4 


221886 


34 


11 


779434 


3484 


999212 


i3 


780222 


3480 


219778 

217680 


33 


781524 


3467 


999205 


i3 


782320 


32 


29 


7836o5 


345i 


999197 
999189 


i3 


784408 


3464 


215592 


3i 


3o 


785675 


343 1 


i3 


786486 


3447 


2i35i4 


3o 


3i 


8-787736 


34i8 


9-999181 


i3 


8-788554 


343 1 


1 1-2 1 1446 


3 


32 


789787 
791828 


3402 


999174 


i3 


790613 


3414 


209387 
207338 


33 


3386 


999166 


i3 


792662 


3399 
3383 


ll 


34 


793859 


3370 


999158 


i3 


794701 


205299 


35 


795881 


3354 


999 1 5o 


i3 


7 9 6 7 3i 


3368 


203269 
201248 


25 


36 


797894 


333 9 
33 2 3 


999142 


i3 


798752 
800763 


3352 


24 


ll 


799897 


999134 


i3 


333 7 


199237 


23 


801892 


33o8 


999126 


i3 


802765 


3322 


197235 


22 


3 9 


803876 


32g3 


9991 18 


i3 


8o4758 


33o7 


195242 


21 


40 


8o5852 


3278 


9991 10 


i3 


806742 


3292 


193258 


20 


4i 


8-807819 


3263 


9-999102 


i3 


8-808717 


3278 


II-IQI283 


8 


42 


809777 


3249 


999094 
999086 


14 


8io683 


3262 


189317 


43 


811726 


3234 


14 


812641 


3248 


187359 


\l 


44 


813667 


32ig 
32o5 


999077 


14 


8i458 9 


3233 


18.5411 


45 


815599 


999069 


14 


816529 


3219 


1 8347 1 


i5 


46 


817522 


3191 


999061 


14 


818461 


3203 


181539 


14 


% 


819436 


3i 77 
3i63 


999053 


14 


820384 


3l9I 


179616 


i3 


82i343 


999044 


14 


822298 


3l77 


177702 


12 


49 


823240 


3i49 
3i35 


999036 


14 


824205 


3i63 


173795 


11 


5o 


825i3o 


999027 


14 


826103 


3i5o 


i 7 38 97 


10 


5i 


8-827011 
828884 


3l22 


9.999019 


14 


8-827992 


3i36 


11-172008 


I 


52 


3io8 


999010 


14 


829874 


3i23 


170126 


53 


830749 


3095 
3o82 


999002 


14 


831748 


3no 


168252 


I 


54 


832607 


998993 


14 


8336i3 


3096 
3o83 


166387 


55 


834456 


3069 


998984 


14 


835471 


164529 


5 


56 


836297 
838i3o 


3o56 


998976 


14 


837321 


3070 


162679 


4 


n 


3 043 


998967 
998958 


i5 


839163 


3o57 


160837 


3 


839956 


3o3o 


i5 


840998 


3o45 


159002 


2 


5 9 


841774 


3017 


998950 


i5 


842825 


3o32 


157175 


I 


60 


843585 


3oco 


998941 


i5 


844644 


3019 


155356 





/ 


Coeine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


93< 


» 












86° 



22 


LOGARITHMIC SINES, 


TANGENTS, ETC 


Table II. 


4° 














175° 


i 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 





8-843585 


3oo5 


9.998941 


i5 


8-844644 


3019 


11-155356 


60 


i 


845387 


2902 
2900 


998932 


i5 


846455 


3007 


153545 


5 Q 


2 


847183 


998923 


i5 


848260 


2995 
2982 


151740 


58 


3 


848971 


2967 
2905 


998914 


i5 


85oo57 


149943 


57 


4 


85o75i 


998905 
998896 
998887 


i5 


85 1 846 


2970 


I48i54 


56 


5 


852525 


2943 


i5 


853628 


2958 


146372 


55 


6 


854291 


2931 


i5 


8554o3 


2946 


144597 


54 


7 


856049 


2919 


998878 


i5 


857171 
858 9 32 


2935 


142829 
141068 


53 


8 


85 7 8oi 


2907 
2896 
2884 


998869 


i5 


2923 


52 


9 


• 85g546 


998860 


:5 


860686 


2911 


i3g3i4 


5i 


10 


861283 


998801 


i5 


862433 


2900 


137567 


5o 


ii 


8-863oi4 


2873 


9-998841 


i5 


8.864173 


2888 


11 -135827 

134094 


49 


12 


864738 


2861 


998832 


i5 


865906 


2877 


48 


i3 


866455 


285o 


998823 


16 


86 7 632 


2866 


132368 


47 


14 


868i65 


283 9 


9 9 88l3 


16 


86 9 3 5 1 


2854 


i3o649 


46 


i5 


869868 


2828 


998804 


16 


871064 


2843 


128 9 36 


45 


16 


87 1 565 


2817 


998793 
998785 


16 


872770 


2832 


127230 


44 


H 


873255 


2806 


16 


874469 


2821 


i2553i 


43 


18 


8 7 4 9 38 


21< £ 

2786 


998776 


16 


876162 


2811 


123838 


42 


19 


876615 


998766 


16 


877849 


2800 


I22l5l 


41 


20 


878285 


2773 


998757 


16 


879529 


2789 


1 20471 


4o 


21 


8 • 879949 


2 7 63 


9.998747 
998738 


16 


8.881202 


2779 
2768 


11-118798 


3 9 


22 


881607 


2752 


16 


882869 


Ii7i3i 


38 


23 


' 883258 


2742 


998728 


16 


88453o 


2758 


1 1 5470 


37 


24 


884903 


2731 


998718 


16 


886i85 


2747 


n38i5 


36 


25 


886542 


2721 


998708 


16 


88 7 833 


2737 


112167 


35 


26 


888174 


271 1 


998699 
998689 


16 


889476 


2727 


iro524 


34 


27 


889801 


2700 


16 


891112 


2717 


108888 


33 


28 


891421 


2690 
2680 


998679 


16 


892742 


2707 


107258 


32 


29 


893035 


998669 


17 


894366 


2697 


io5634 


3i 


3o 


894643 


2670 


998659 


17 


895984 


2687 


104016 


3o 


3i 


8-896246 


2660 


9-998649 


17 


8-897596 


2677 


11 102404 


11 


32 


897842 


265i 


998639 


n 


899203 


2667 


100797 


33 


899432 


2641 


998629 


17 


900803 


2658 


099197 


*1 


34 


901017 


263i 


998619 


n 


902398 


2648 


097602 


26 


35 


902596 


2622 


'998609 


n 


903987 


2638 


096013 


25 


36 


904169 


2612 


9985o 9 
998089 
998578 


n 


905370 


2629 


094430 


24 


37 


905736 


26o3 


n 


907147 


2620 


092853 


23 


38 


907297 


25o3 


17 


90S719 


2610 


091281 
089715 
o88i54 


22 


39 


908853 


2584 


998568 


n 


910280 


2601 


21 


4o 


910404 


2573 


998558 


17 


911846 


2592 


20 


4i 


8-91 1949 


2566 


9-998548 


17 


8-9i34oi 


2583 


11 -086599 


\l 


42 


913488 


2556 


998537 


n 


914921 


2574 


085049 


43 


9i5o22 


2547 


998527 


n 


916495 


2565 


oS35oo 


17 


44 


9i655o 


2538 


9985 1 6 


18 


918034 


2556 


081966 


16 


45 


918073 


2D29 


9g85o6 


18 


919568 


2547 


080432 


i5 


46 


919591 


2520 


998495 
998480 


18 


921096 


2538 


078904 
077381 


14 


47 


921103 


25l2 


18 


922619 
924i36 


253o 


i3 


48 


922610 


25o3 


998474 


18 


2521 


075864 


12 


49 


924112 


24q4 


998464 


18 


925649 


2012 


07435i 


11 


5o 


925609 


2486 


99S453 


18 


927156 


2003 


072844 


10 


5i 


8-927100 


2477 


9-998442 


18 


8-928658 


2490 


11-071342 


\ 


52 


928587 


2469 


998431 


18 


93o'55 


2486 


069845 
068353 


53 


930068 


2460 


998421 


18 


931647 


2473 


I 


54 


93 1 544 


2452 


998410 


18 


9 33 1 34 


2470 


066866 


55 


933oi 5 


2443 


9983^ 


18 


934616 


2461 


o65384 


5 


56 


934481 


2435 


18 


936093 


2453 


063907 


4 


57 


935942 
937398 


2427 


998377 


18 


937565 


2445 


062435 


3 


58 


2419 


998366 


18 


939032 


2437 


060968 


2 


5o 


93885o 


241 1 


998355 


18 


940494 


243o 


059006 


1 


6o 


940296 


24o3 


098344 


18 


941952 


2421 


00S048 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


1 


94- 














85° 



Table II. LOGARITHMIC SINES 


, TANGENTS, ETC. 


28 


5° 














174° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


f 


o 


8-940296 
941738 


24o3 


9-998344 


19 


8-941952 


2421 


n-o58o48 


60 


i 


23 9 4 
238 7 


998333 


19 


943404 


24i3 


056596 


U 


2 


943174 


998322 


»9 


944852 


24o5 


o55i48 


3 


944606 


23 79 


9983 1 1 


19 


946295 


23 97 


o537o5 


57 


4 


946o34 


2371 


998300 


19 


947734 


2390 


002266 


56 


5 


947456 


2363 


998289 


i<; 


949168 


2382 


o5o832 


55 


6 


948874 


2355 


998277 


19 


950597 


23 7 4 


049403 


54 


I 


950287 


2348 


998266 


19 


952021 


2366 


04797*9 
046559 


53 


951696 


234o 


998255 


19 


953441 


236o 


5a 


9 


953lOO 


2332 


998243 


19 


9 54856 


235i 


045 1 44 


5i 


IO 


954499 


2325 


998232 


19 


956267 


2344 


043733 


30 


II 


8.955894 


23i7 


9-998220 


19 


8-957674 


233 7 


11-042326 


% 


12 


957284 
958670 


23lO 


998209 


19 


959075 


2329 

2323 


040925 
039027 


i3 


23o2 


998197 


19 


960473 


47 


14 


96oo52 


2295 


998186 


*9 


961866 


23i4 


o38i34 


46 


i5 


961429 


2288 


998174 


19 


963255 


2307 


036745 


45 


16 


962801 


2280 


998163 


19 


964639 


23oo 


o3536i 


44 


\l 


964170 


2273 


998l5l 


19 


966019 


2293 


033981 


43 


965534 


2266 


998139 
998128 


20 


967394 


2286 


032606 


42 


19 


966893 


2259 


20 


968766 


2279 


o3i234 


4i 


20 


968249 


2252 


998116 


20 


97oi33 


2271 


029867 


4o 


21 


8-969600 


2244 


9-998104 


20 


8-971496 


2265 


u-0285o4 


39 


22 


970947 


2238 


998092 


20 


972855 


2257 


027145 


38 


23 


972289 
973628 


223l 


998080 


20 


974209 


225l 


025791 


I 1 


24 


2224 


9 9 8o68 


20 


97556o 


2244 


024440 


36 


25 


974962 


2217 


998056 


20 


976906 


2237 


023094 


35 


26 


976293 


2210 


998044 


20 


978248 . 


223o 


021762 


34 


S 


977619 


2203 


998032 


20 


979586 


2223 


020414 


33 


978941 


2197 


998020 


20 


980921 


2217 


019079 


32 


29 


980259 
981573 


219O 


998008 


20 


982251 


22IO 


017749 
016423 


3i 


3o 


2i83 


997996 


20 


983577 


2204 


3o 


3i 


8-982883 


2177 


9.997984 


20 


8-984899 


2197 


n«oi5ioi 


3 


32 


984189 


2170 


997972 


20 


986217 


2191 


013783 


33 


985491 
986789 


2i63 


997959 


20 


987532 


2l84 


012468 


27 


34 


2157 


997947 


20 


988842 


2178 


oiii58 


26 


35 


988083 


2i5o 


997935 


21 


99 OI 49 


2171 


ooq85i 
008549 


25 


36 


989 3 74 


2144 


997922 


21 


99i45i 


2i65 


24 


12 


990660 


2i38 


997910 


21 


992750 


2i58 


007250 


23 


991943 


2l3l 


?$8 


21 


994045 


2 1 52 


000955 


22 


39 


993222 


2125 


21 


995337 


2146 


004663 


21 


4o 


994497 


21 19 


997872 


21 


996624 


2140 


003376 


20 


41 


8-995768 


21 12 


9-997860 


21 


8-997908 


2i34 


11 -002092 


\% 


42 


997036 


2106 


997835 


21 


999188 


2127 


000812 


43 


998299 


2100 


21 


9 - 000465 


2121 


10-999535 
998262 


17 


44 


999560 


2094 


997822 


21 


001738 


2Il5 


16 


45 


9-000816 


20»7 


997809 


21 


003007 


2109 
2io3 


996993 


i5 


46 


002069 
oo33i8 


2082 


997797 


21 


004272 


995728 


14 


47 


2076 


997784 


21 


oo5534 


2097 


994466 


i3 


48 


oo4563 


207O 


997771 


21 


006792 


2091 

2o85 


993208 


12 


4 9 


oo58o5 


2064 


997758 


21 


008047 
009298 


991953 


11 


5o 


007044 


2o58 


997745 


21 


2080 


990702 


10 


5i 


9-008278 


2052 


9.997732 


21 


9- oio546 


2074 


1 a- 989454 
988210 


I 


52 


009510 


2046 


9977 1 9 


21 


01 1790 


2068 


53 


010737 


2040 


997706 


21 


oi3o3i 


2062 


986969 


7 


54 


01 1962 


2o34 


997693 


22 


014268 


2o56 


985732 


6 


55 


0i3i82 


2029 


997680 


22 


oi55o2 


2o5i 


984498 


5 


56 


014400 


2023 


997667 


22 


016732 


2045 


9 83268 


4 


tl 


0i56i3 


2017 


997654 


22 


017959 


2040 


982041 


3 


016824 


2012 


997641 


22 


019183 


2o33 


980817 


2 


5 9 


oi8o3i 


2006 


997628 


22 


02o4o3 


2028 


979507 
978380 


1 


6o 


019235 


2000 


997614 


22 


021620 


2023 




I 


i 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


95 s 
















84° 



24 


LOGARITHMIC SINES, 


TANGENTS, ETC 


Table 


II. 


6° 














173° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 


o 


9-OI9235 


2000 


9-9976I4 


22 


9-021620 


2023 


10-978380 


60 


I 


020435 


I9o5 


007601 


22 


022834 


2017 


977166 


u 


2 


021632 


I989 


997688 


22 


024044 


201 1 


973g56 


3 


022825 


1984 


997574 


22 


02525l 


2006 


974749 


57 


4 


024oi6 


I978 


997661 


22 


026455 


2000 


973543 


06 


5 


025203 


I 97 3 


997547 


22 


027655 


1995 


972345 


55 


6 


026386 


I967 


997534 


23 


028852 


I90O 
I 9 85 


971 U8 


54 


I 


027667 
028744 


I962 


997520 


23 


o3oo46 


969954 
968763 


53 


I 9 5 7 


997507 


23 


o3i237 


I979 


52 


9 


029918 


I9DI 


9974o3 
997480 


23 


o32425 


1974 


967575 


5i 


10 


031089 


1947 


23 


033609 


I969 


966391 


5o 


ii 


9«o32257 


1941 


9-997466 


23 


9-034791 


1964 


IO-965209 


a 


12 


o3342i 


1936 


997432 


23 


035969 


i 9 58 


96403 1 


i3 


o34582 


1930 


997439 
997423 


23 


037144 
o383i6 


1933 


962856 


47 


U 


035741 


1925 


23 


1948 


961684 


46 


i5 


o368 9 6 


I920 


99741 1 


23 


039483 


1943 


96o5i5 


45 


16 


o38o48 


I9l5 


997397 
997383 


23 


o4o65i 


i 9 38 


959349 

958187 


44 


12 


039197 


I91O 


23 


o4i8i3 


i 9 33 


43 


o4o342 


IOOD 


997369 


23 


042973 


1928 


957027 


42 


19 


041485 


i 8 99 


997355 


23 


044 1 3o 


1923 


906870 


4i 


20 


042620 


1894 


997341 


23 


046284 


1918 


934716 


40 


21 


9-043762 
044895 


ig|9 


9.997327 


24 


9-046434 


1913 


10-953566 


39 


22 


1884 


9973 1 3 


24 


047682 


1908 


962418 


38 


23 


046026 


l8 n 


997299 


24 


048727 


1903 
1898 


901273 


37 


24 


047i54 


i8 7 5 


997283 


24 


049860 


9Doi3i 


36 


25 


048279 


1870 


997271 


24 


oSiooo 


1893 
1889 


948992 


35 


26 


049400 


1 865 


997237 


24 


o52i44 


947836 


34 


2 


o5o5io 
o5i635 


i860 


997242 


24 


053277 


1884 


946723 


33 


i855 


997228 


24 


064407 
o55535 


1879 


945393 


32 


29 


952749 


i85o 


997214 


24 


1874 


944465 


3i 


3o 


o53859 


i845 


997199 


24 


o5665g 


1870 


943341 


3o 


3i 


9-054966 


1 841 


9-997183 


24 


9-057781 


1 865 


10-942219 


3 


32 


056071 


1 836 


997170 


24 


o58ooo 


1869 


941 100 


33 


057172 
058271 


i83i 


997 1 56 


24 


060016 


i853 


939984 
938870 


11 


34 


1827 


997141 


24 


o6u3o 


i85i 


35 


069367 


1822 


997127 


24 


062240 


1846 


937760 


25 


36 


060460 


1817 


997112 


24 


063348 


1842 


936652 


24 


37 


o6i55i 


i8i3 


997098 


24 


064453 


i83 7 


935547 


23 


38 


062639 


1808 


997083 


25 


065556 


1 833 


934444 


22 


39 


063724 
064806 


1804 


997068 


25 


o66655 


1828 


933345 


21 


4o 


1799 


997053 


25 


067762 


1824 


932248 


20 


4i 


9-o65885 


1794 


9-997039 


25 


9-068846 


1819 


io-93ii54 


!3 


42 


066962 


1790 


997024 


25 


069938 


l8l3 


930062 


43 


o68o36 


1786 


997009 


25 


071027 


1810 


928973 
927887 
926803 


3 


44 


069107 


1781 


996994 


23 


072113 


1806 


45 


070176 


1777 


996979 


25 


073197 
074278 


1802 


i5 


46 


071242 


1772 


996964 


25 


1797 


926722 


14 


% 


072306 


1768 


996949 


25 


075356 


I7g3 
1789 


924644 


i3 


073366 


1763 


996934 


25 


076432 


923568 


12 


40 


074424 


17D9 


996919 


25 


077605 


1784 


922495 


11 


5o 


075480 


1753 


996904 


23 


078376 


1780 


921424 


10 


5i 


9-076533 


1760 


9-996889 


25 


9-070644 


1776 


io«92o356 


I 


52 


077583 


1746 


996874 


25 


080710 


1772 


919290 
91S227 


53 


078631 


1742 


996858 


25 


081773 


1767 


I 


54 


079676 


i 7 38 


996843 


25 


o82833 


1763 


917167 


55 


080719 


i 7 33 


996828 


25 


083891 


1739 


916100 
9i5o53 


5 


56 


081759 


1729 


99681a 


26 


084947 


1735 


4 


n 


082797 
o83832 


1723 


996707 


26 


086000 


1731 


914300 


3 


1721 


996782 


26 


087050 
088098 


1747 


912950 


2 


5o 


084864 


nn 


996766 


26 


1743 


91 1902 
9iob56 


1 


6o 


086894 


I7i3 


996731 


26 


089144 


1738 



1 


/ 


Cosine. 


D. 


Sine. 


* 


Cotang. 


a 


Tang. 


96 c 














1 


>3° 



Table II. LOGARITHMIC SINES 


, TANGENTS, ETC. 25 


7° 














172° 


o 


Sine. 


D. 


Cosine. 


D 


Tang. 


D. 


Cotang. | / 


9 -085894 


I7l3 


9-996751 


26 


9-089144 


1738 


lo-9io856 


60 


i 


086922 


1709 


996735 


26 


090187 


1734 


90o8i3 
908772 


\n 


2 


087947 


1704 


996720 


26 


091228 


1730 


3 


088970 


1700 


996704 


26 


092266 


1727 


907734 5 7 


4 


089990 


1696 


996688 


26 


093302 


1722 


906698 55 


5 


091008 


1692 
1688 
1684 


996673 


26 


094336 


1719 


9o5664 55 


6 


092024 
093o37 


996657 
996641 


26 
26 


096367 
O96395 


1713 
1711 , 


904633 54 
9o36o5 53 


8 


094047 


1680 


996625 


26 


097422 
098446 


1707 


902578 52 


9 


095o56 


1676 


996610 


26 


1703 


90i554 5i 


IO 


096062 


l6 7 3 


996594 


26 


099468 


1699 


900532 5o 


ii 


9-097065 


1668 


9-996578 


27 


9.IOO487 


i6 9 5 


10-899513 49 
898496 48 
897481 47 


12 


098066 


1 665 


996562 


27 


ioi5o4 


1691 
1687 


i3 


099065 


1661 


996546 


27 


102519 


U 


100062 


i65 7 
1 653 


996530 


27 


io3532 


1684 


896468 . 46 


i5 


ioio56 


9965l4 


27 


104542 


1680 


8 9 5458 45 


16 


102048 


1649 
1645 


996498 
996482 


27 


io555o 


1676 


894450 


44 


\l 


io3o37 
104025 


27 


io6556 


1672 


893444 


43 


1641 


996465 


27 


107559 


1669 
1 665 


892441 


42 


19 


io5oio 


i638 


996440 
996433 


27 


io856o 


891440 


41 


20 


105992 


1 634 


27 


109559 


1661 


890441 


40 


21 


9« 106973 
107901 


i63o 


9-996417 


27 


9.iio556 


1658 


10-889444 


u 


22 


1627 


996400 


27 


iii55i 


i654 


888449 


23 


108927 


1623 


996384 


27 


1 1 2543 


i65o 


887457 


n 


24 


1 0990 1 


1619 
1616 


9 9 6368 


27 


1 i3533 


1 646 


886467 


25 


1 1 0873 


996351 


27 


ii452i 


1643 


885479 
884493 


35 


26 


1 1 1842 


1612 


996335 


27 


1 1 5507 


1639 


34 


s 


1 1 2809 


1608 


996318 


27 


116491 


1636 


8835o 9 
882528 


33 


1 13774 


i6o5 


996302 


28 


1 17472 • 
1 i8452 


i632 


32 


29 


1 14737 


1601 


996285 


28 


1629 
1625 


88i548 


3i 


3o 


n56 9 8 


i5 97 


996269 


28 


1 19429 


88o5 7 i 


3o 


3i 


9-ii6656 


1594 


9-996252 


28 


9.120404 


1622 


10-879596 


S 


32 


1 17613 

1 1 8567 


1590 
1 58 7 
1 583 


996235 


28 


121377 


1618 


878623 


33 


996219 


28 


122348 


i6i5 


877652 


11 


34 


119519 


996202 


28 


123317 


1611 


8 7 6683 


35 


120469 


i58o 


996185 


28 


124284 


1607 


875716 


25 


36 


121417 


i5 7 6 


996168 


28 


125249 


1604 


874751 


24 


37 


122362 


i5 7 3 


996151 


28 


1 262 1 1 


1601 


873789 
872828 


23 


38 


i233o6 


1569 


996134 


28 


127172 


1 597 


22 


3 9 


124248 


1 566 


9961 17 


28 


I28i3o 


1594 


871870 


21 


4o 


I25i8 7 


1 562 


996100 


28 


129087 


1591 


870913 


20 


4i 


9-126125 


i559 


9-996083 


29 


9-i3oo4i 


i58 7 


1 • 869959 


;g 


42 


127060 


i556 


996066 


29 


1 30994 


1 584 


869006 
868o56 


43 


127993 


i552 


996049 


29 


131944 
i328 9 3 


i58i 


u 


44 


128923 


1 549 
1 545 


996032 


29 


1 577 


867107 


45 


129854 


996015 


29 


1 3383g 


1 574 


866161 


i5 


46 


130781 


1 542 


995998 


29 


134784 


i5 7 i 


8652i6 


14 


% 


131706 


1539 
i535 


993980 


29 


i35 7 26 


1 56 7 


864274 


i3 


i3263o 


995963 


29 


1 3666 7 


1 564 


863333 


12 


49 


i3355i 


i532 


995946 


29 


i3 7 6o5 

138542 


i56i 


862395 
861458 


11 


5o 


134470 


1529 


995928 


29 


i558 


10 


5i 


9-135387 


i525 


9.99501 1 
990894 


29 


9- 139476 


1555 


io-86o524 


I 


52 


i363o3 


l522 


29 


140409 


i55i 


859591 
85866o 


53 


137216 

i38i28 


i5i.g 


99 58 7 6 


29 


1 41 34o 


1 548 


I 


54 


i5i6 


995859 


29 


142269 


1 545 


85 7 73i 


55 


139037 


l5l2 


995841 


29 


143196 


1 542 


8568o4 


5 


56 


139944 


1 509 


995823 


29 


144121 


1539 
1535 


855879 


4 


u 


i4o85o 


i5o6 


995806 


29 


i45o44 


854956 


3 


I4I754 


i5o3 


995788 


29 


145966 

146885 


1532 


854o34 


2 


59 


142655 


i5oo 


995771 


29 


1529 


853n5 


1 


60 


U3555 


1496 


995753 


29 


147803 


l520 


852197 





/ 


Canine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


97° 














82° 



26 


LOGARITHMIC SINES, 


TANGENTS, ETC 


Table II. 


8° 














171° 


t 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


1 





9-143555 


1496 


9-995753 


3o 


9«l478o3 


i526 


10-852197 

85i28a 


60 


i 


144453 


1493 


995735 


3o 


148718 


i523 


% 


2 


145349 
146243 


14Q0 
I4«7 


995717 


3o 


I4g632 


1 520 


85o368 


3 


995699 


3o 


i5o544 


i5i7 


849456 
848546 


57 


4 


I47i36 


1484 


995681 


3o 


i5i454 


i5i4 


56 


5 


148026 


I48l 


995664 


3o 


152363 


i5ii 


847637 


55 


6 


148013 
149802 


1478 


995646 


3o 


153269 


i5o8 


846731 


54 


I 


1475 


995628 


3o 


i54i74 


i5o5 


845826 


53 


100686 


1472 


995610 


3o 


155077 


l5o2 


844923 


52 


9 


1 5 1 569 


1469 


995591 


3o 


155978 


1499 


844022 


5i 


10 


i5245i 


1466 


995573 


3o 


1 568 77 


1496 


843i23 


5o 


ii 


9'i5333o 


1463 


9-995555 


3o 


9.157775 


U93 


10-842225 


% 


12 


1 54208 


1460 


995537 


3o 


1 5867 1 


1 490 


841329 
840435 


i3 


i55o83 


1457 


9 9 55 ig 


3o 


159565 


1487 


47 


14 


155937 
1 5683o 


1454 


9955oi 


3i 


160457 


1484 


839543 


46 


i5 


i45i • 


995482 


3i 


161347 


1481 


838653 


45 


16 


157700 


1448 


995464 


3i 


I&2236 


U79 


837764 


44 


H 


i5856c) 


1445 


995446 


3i 


i63i 2 3 


U76 


836877 


43 


i8 


1 59433 


1442 


995427 


3i 


164008 


1473 


835992 


42 


19 


i6o3oi 


1439 


995409 


3i 


164892 


1470 


835 1 oS 


4i 


20 


161164 


1436 


995390 


3i 


165774 


1467 


834226 


4o 


21 


9-162025 


1433 


9-995372 


3i 


9.166654 


1464 


10-833346 


M 


22 


162885 


i43o 


995353 


3i 


167532 


1461 


832468 


.23 


163743 


1427 


995334 


3i 


168409 


1458 


83i5 9 i 


37 


24 


164600 


1424 


9953i6 


3i 


169284 


1455 


830716 


36 


25 


165454 


1422 


995297 


3i 


170157 


1453 


829843 


35 


26 


1 663 07 


1419 
1416 


995278 


3i 


171029 


i45o 


828971 


34 


3 


167159 
168008 


995260 


3i 


171899 


1447 


828101 


33 


I4i3 


995241 


32 


172767 


1444 


827233 


32 


29 


168856 


1410 


995222 


32 


173634 


1442 


826366 


3i 


3o 


169702 


1407 


995203 


32 


174499 


1439 


8255oi 


3o 


3i 


9-I70547 


i4o5 


9-995184 


32 


9.175362 


1436 


10-824638 


3 


32 


171389 


1402 


995 1 65 


32 


176224 


1433 


823776 


33 


172230 


1 399 


995146 


32 


177084 


143 1 


822916 


27 


34 


173070 


1396 


9 9 5 1 27 
995108 


32 


177942 


1428 


822058 


26 


35 


173908 


i3 9 4 


32 


178799 


1425 


821201 


25 


36 


174744 


i3oi 


995089 


32 


179653 


1423 


820345 


24 


3? 


175578 


1 388 


995070 


32 


i8o5o8 


1420 


819492 
818640 


23 


38 


176411 


1 386 


993o5i 


32 


i8i36o 


1417 


22 


3 9 


177242 


1 383 


995o32 


32 


182211 


I4i5 


817789 


21 


40 


178072 


1 38c 


995o 1 3 


32 


183039 


1412 


816941 


20 


4i 


9-178900 


i3 77 


9.994993 


32 


9.183907 


1409 


10-816093 


:i 


42 


179726 


i3 7 4 


994974 


32 


184752 


1407 


813248 


43 


i8o55i 


1372 


994935 


32 


185597 
186439 


1404 


814403 


17 


44 


181374 


1369 


994935 


32 


1402 


8i356i 


16 


45 


182196 


1 366 


994916 
994896 


33 


187280 


1399 


812720 


i5 


46 


i83oi6 


1 364 


33 


188120 


1396 


81 1880 


14 


% 


183834 


i36i 


994877 
994857 
994838 


33 


i88 9 58 


i3q3 


81 1042 


i3 


1 8465 1 


i359 


33 


189794 


i3oi 

1 38 9 


810206 


12 


49 


185466 


i356 


33 


190629 


80937: 
8o8538 


11 


5o 


186280 


i353 


994818 


33 


191462 


i386 


10 


5i 


9-187092 


i35i 


9.994798 


33 


9-192294 


1384 


10-807706 


8 


52 


187903 
188712 


1 348 


994779 
994759 


33 


193124 


i38i 


806876 


53 


1 346 


33 


193953 


1 3 79 


806047 


I 


54 


189519 


1 343 


994739 


33 


194780 


1376 


805220 


55 


190323 


i34i 


994720 


33 


195606 


1374 


804394 


5 


56 


191130 


1338 


994700 


33 


196430 


i3 7 i 


803570 


4 


n 


191933 


1336 


994680 


33 


197253 


1 369 


802747 
801920 


J 


192734 


i333 


994660 


33 


198074 


1 366 


2 


59 


193534 


i33o 


994640 


33 


198894 


1 364 


801 106 


I 


60 


194332 


1328 


994620 


33 


199713 


i36i 


800287 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


t 


98° 














81° 



Table II. LOGARITHMIC SINES 


TANGENTS, ETC. 27 


9° 














1*0° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


t 


o 


9-I94332 


i328 


9-994620 


33 


9-I997I3 


i36i 


10-800287 


60 


i 


I95129 


1326 


994600 


33 


20052O 

201345 


1359 


799471 
7 9 8655 


to 


2 


I9592D 


1323 


994080 


33 


i356 


3 


I96719 


l32I 


994560 


34 


202159 


i354 


797841 


57 


4 


19761 1 


i3i8 


994540 


34 


202971 


i352 


797020 
796218 


56 


5 


198302 


i3i6 


994519 


34 


203782 


1 349 


55 


6 


I 9909 I 


i3i3 


994499 


34 


204592 


1 347 


795408 


54 


7 


199879 


i3ii 


994479 


34 


2o54oo 


1 345 


794600 


53 


8 


200666 


i3o8 


99445Q 


34 


206207 


1 342 


7 9 3 79 3 


5a 


9 


2oi45i 


i3o6 


994438 


34 


20701$ 


1 34o 


792987 


5i 


10 


202234 


i3o4 


994418 


34 


207817 


1338 


792183 


5o 


ii 


9.203017 


i3oi 


9-994398 


34 


9-208619 


i335 


io-79i38i 


3 


12 


203797 


1299 


994377 


34 


209420 


1333 


790580 


i3 


204577 


1296 


994357 


34 


210220 


i33i 


789780 
788982 


47 


i4 


2o5354 


1294 


994336 


34 


2IIOI8 


1328 


46 


i5 


2o6i3i 


1292 


9943l6 


34 


2ii8i5 


. i326 


788185 


45 


16 

12 


206906 
207679 


1289 

1287 


994295 
994274 


34 
35 


212611 
2i34o5 


i324 

1321 


787389 
7865 9 5 


44 
43 


208452 


1285 


994254 


35 


214198 
214989 


i3 19 


785802 


42 


19 


209222 


1282 


994233 


35 


1317 


785ou 


4i 


20 


209992 


1280 


994212 


35 


216780 


i3i5 


784220 


4o 


21 


9-210760 


1278 


9-994191 


35 


9-2i6568 


I3l2 


10-783432 


39 


22 


21 1626 


1275 


994171 


35 


217356 


i3io 


782644 


38 


23 


212291 


1273 


994 1 5o 


35 


218142 


i3o8 


781868 


37 


24 


2i3o55 


1271 


994129 
994108 


35 


218926 


i3o5 


781074 


36 


25 


2i38i8 


1268 


35 


219710 


i3o3 


780290 


35 


26 


214579 
2i5338 


1266 


994087 


35 


220492 


i3oi 


779508 
778728 


34 


11 


1264 


994066 


35 


221272 


1299 


33 


216097 


1261 


994045 


35 


222052 


1297 


777948 


32 


29 


2i6854 


1259 


994024 


35 


222830 


1294 


777170 


3i 


3o 


217609 


1257 


994003 


35 


223607 


1292 


776393 


3o 


3i 


9-2i8363 


1255 


9-993982 


35 


9-224382 


1290 


10-775618 


20 


32 


219116 


1253 


993960 


35 


225i56 


1288 


774844 


28 


33 


219868 


1250 


993939 
993918 


35 


225929 


1286 


774071 


27 


34 


220618 


1248 


35 


226700 


1284 


7733oo 


26 


35 


221367 


1246 


99 38 9 7 


36 


227471 


1281 


772529 


25 


36 


222II5 


1244 


993875 


36 


228239 


1279 


771761 


24 


37 


222861 


1242 


993854 


36 


229007 


1277 


770993 


23 


38 


2236o6 


1239 


993832 


36 


229773 


1273 


770227 


22 


39 


224349 


1237 


9938 1 1 


36 


23o539 


1273 


769461 


21 


4o 


225092 


1235 


993789 


36 


23l302 


1271 


768698 


20 


4i 


9-225833 


1233 


9-993768 


36 


9-232o65 


1269 


10-767935 


\l 


42 


226573 


1 23 1 


993746 


36 


232826 


1267 


767174 


43 


227311 


1228 


993725 


36 


233586 


1265 


766414 


17 


44 


228048 


1226 


993703 


36 


234345 


1262 


765655 


16 


45 


228784 


1224 


993681 


36 


235io3 


1260 


764897 


i5 


46 


229518 


1222 


993660 


36 


235859 


1258 


764141 


14 


4 1 


230252 


1220 


993638 


36 


2366i4 


1256 


763386 


i3 


4$ 


230984 


I2l8 


993616 


36 


237368 


1254 


762632 


12 


49 


231715 


I2l6 


993594 


37 


238120 


1252 


761880 


11 


5o 


232444 


1214 


993572 


37 


2388 7 2 


125o 


761128 


10 


5i 


9-233i72 


1212 


9-99355o 


I 1 


9-239622 


1248 


10-760378 


I 


52 


233899 


1209 


993528 


?7 


240371 


1246 


759629 


53 


234625 


1207 


9935o6 


37 


241118 


1244 


758882 


I 


54 


235340 
236073 


1205 


993484 


37 


241 865 


1242 


758i35 


55 


1203 


993462 


37 


242610 


1240 


757390 


5 


56 


236795 


1201 


993440 


37 


243354 


1238 


756646 


4 


u 


23 7 5i5 


II99 


993418 


37 


244097 
244839 


1236 


755903 


3 


238235 


1 197 


993396 


37 


1234 


755i6i 


2 


5 9 


238g53 


1 1 9 5 


993374 


37 


245579 


1232 


764421 


I 


60 


239670 


1193 


99335i 


37 


246319 


I23o 


75368i 





Cosine, 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


1 


99 c 


> 












80 c 



28 


LOGARITHMIC SINES 


TANGENTS, ETC. Table II. 


10 c 














169° 


/ 


S : ne. 


D. 


Cosine. 


D. 


Tang. 


1 D - 


Cotang. 


1 





9-239670 


1 193 


9-99335I 


37 


9-246319 


1230 


io-75368i 


60 


i 


24o386 


1 189 


993329 


I 1 


247057 


1228 


752943 


a 


2 


241 101 


993307 


? 7 


247794 


1226 


752206 


3 


241814 


H87 


993284 


37 


24853o 


1224 


731470 


5 7 


4 


242526 


n85 


993262 


37 


249264 


1222 


750736 


56 


5 


243237 


n83 


993240 


37 


249998 


1220 


750002 


55 


6 


243947 


1181 


993217 


38 


25o73o 


I2l8 


740270 


54 


I 


244656 


1 179 


993193 


38 


25i46i 


1217 


748539 


53 


24-363 


1177 


993172 


38 


252191 


12l5 


747809 


52 


9 


246069 


1175 


993149 


38 


252920 


1213 


747080 


5i 


10 


246770 


n 7 3 


9 9 3l2 7 


38 


253648 


I2II 


746352 


5o 


ii 


9-247478 


1171 


9-993104 


38 


9-204374 


I209 


10.745626 


% 


12 


248181 


1 169 


993o8l 


38 


255ioo 


1207 


744900 


i3 


248883 


1 167 


993o59 


38 


255824 


I2o5 


744176 


47 


U 


249583 


1163 


993o36 


38 


256547 


I2o3 


743453 


46 


i5 


250282 


n63 . 


993oi3 


38 


257269 


1201 


74273i 


45 


16 


250980 


1161 


992990 


38 


257990 


I200 


742010 


44 


n 


251677 
252373 


1 1 59 
n58 


992967 


38 


258710 


II98 


741290 


43 


18 


992944 


38 


259429 


II96 


740571 


42 


19 


253o67 


1 1 56 


992921 


38 


260146 


1 1 9 4 


739854 


4i 


20 


253761 


n54 


992898 


38 


26o863 


II92 


739137 


4o 


21 


9-254453 


Il52 


9-992875 


38 


9-261578 


1 190 
I189 


10-738422 


3 9 


22 


255i44 


n5o 


992852 


38 


262292 


737708 


38 


23 


255834 


1 148 


992829 
992806 


39 


263oo5 


U87 


736995 
736283 


37 


24 


• 256523 


1 146 


3 9 


263717 
264428 


n85 


36 


25 


257211 


II44 


992783 


39 


n83 


735572 


35 


26 


257898 
258583 


1142 


992759 


3 9 


265i38 


1181 


734862 


34 


3 


1 141 


992736 


3 9 


265847 


\\]l 


734i53 


33 


259268 


1139 


992713 


3 9 


266555 


733445 


32 


29 


259951 


n3 7 


992690 


39 


267261 


1176 


732 7 3o 
732033 


3i 


3o 


26o633 


u35 


992666 


3 9 


267967 


ii74 


3o 


3i 


9-26i3i4 


n33 


9-992643 


39 


9.268671 


1172 


10.731329 


ll 


32 


261994 


ii3i 


992619 


3 9 


269373 


1 1 70 


73o623 


33 


265673 


ii3o 


992596 


3 9 


270077 


1 169 


729923 


27 


34 


26335i 


1128 


992372 


39 


270779 


1 167 
u65 


729221 

728521 


26 


35 


264027 


1126 


992349 


2° 


271479 
272178 


25 


36 


264703 


1124 


992323 


3 9 


1 164 


727822 


24 


ll 


265377 


1 1 22 


9925oi 


3 9 


272876 


1162 


727124 


23 


266o5 1 


1 1 20 


99 2 478 


40 


273573 


1 160 


726427 


22 


3 9 


266723 


1119 


99 2 434 


40 


274269 


n58 


723731 


21 


4o 


267395 


1117 


992430 


40 


274964 


1137 


725o36 


20 


4i 


9.268065 


iii5 


9-992406 


4o 


9-275658 


n55 


10-724342 


3 


42 


268734 


11 13 


992382 


40 


27635i 


n53 


723649 


43 


269402 


HIT 


992359 


4o 


277043 


n5i 


722957 


17 


44 


270069 


I I 10 


992333 


40 


277734 


n5o 


722266 


16 


45 


270730 


II08 


9923 1 1 


40 


278424 


1 148 


721576 


i5 


46 


27 1 400 


1 106 


992287 


40 


2791 i3 


1 147 


720887 


14 


a 


272064 


no5 


992263 


40 


279801 


1143 


720199 


i3 


272726 


no3 


992239 


40 


280488 


1 143 


7IODI2 

718826 


12 


49 


2 7 3388 


1101 


992214 


40 


281174 


1141 


11 


5o 


274049 


1099 


992190 


40 


28i858 


1140 


718142 


10 


5i 


9.274768 


1098 


9-992166 


40 


9-282542 


n33 


10-717458 


I 


52 


275367 


1096 


992142 


40 


283225 


u36 


716775 


53 


276025 


1094 


992118 


41 


283907 


n35 


716093 


I 


54 


276681 


1092 


992093 


41 


284388 


Ii33 


713412 


55 


277337 


1091 
1089 


992069 


41 


2S5268 


n3i 


714732 


5 


56 


277991 


992044 


41 


285947 


n3o 


714053 


4 


n 


278643 


1087 


992020 


41 


286624 


112S 


713376 


3 


279297 
279948 
280399 


1086 


991996 


41 


287301 


1126 


712690 

712023 


2 


5 9 


1084 


991971 


4i 


287977 


1125 


1 


6o 


1082 


991947 


4i 


288652 


1123 


71 1348 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


IOC 















79° 



Table II. LOGARITHMIC SINES, TANGENTS, ETC. 


29 


11° 












168° 


/ 


Sine. 


D. 


Cosine. 


! D - 


Tang. 


D. 


Cotang. 


r 





9.280599 
281248 


1082 


9.991947 


41 


9-288652 


1 1 23 


IO-7II348 


60 


i 


1 081 


991922 
991807^ 
99 1 873 


41 


289326 


1122 


710674 


U 


2 


281897 


1079 


41 


289999 


1 1 20 


710001 


3 


282544 


1077 
1076 


41 


290671 


IIl8 


709329 
708658 


5 7 


4 


283190 
283836 


99 1 848 


41 


291342 


III7 


56 


5 


1074 


991823 


a I 


292013 


iii5 


707987 
707318 


55 


6 


284480 


1072 


99 l 799 


41 


292682 


1114 


54 


I 


285i24 


IO71 


99 J 774 


42 


29335o 


11 12 


7o665o 


53 


285 7 66 


I069 


991749 


42 


294017 


IIII 


7o5o83 
7o53i6 


52 


9 


286408 


1067 


991724 


42 


294684 


1 109 


5i 


10 


287048 


1066 


991699 


42 


296349 


1 107 


70465 1 


5o 


ii 


9.287688 
288326 


1064 


9.991674 


42 


9-296013 


1 106 


10-703987 


% 


12 


io63 


99 1 649 


42 


296677 


1 104 


703323 


i3 


288964 


1061 


991624 


42 


297339 
298001 


no3 


702661 


% 


i4 


289600 


1059 
io58 


991599 


42 


IIOI 


701999 
7oi338 


i5 


290236 


991 574 


42 


298662 


1 100 


45 


16 


290870 


io56 


99 1 549 


42 


299322 


1098 


700678 


44 


n 


291504 


io54 


991524 


42 


299980 


1096 


700020 


43 


18 


292137 


io53 


991498 


42 


3oo638 


1095 


699362 
698705 


42 


*9 


292768 


io5i 


991473 


42 


3oi2o5 
301961 


1093 


4i 


20 


293399 


io5o 


991448 


42 


1092 


698049 


4o 


21 


9*294029 
294658 


1048 


9-991422 


42 


9.302607 


1090 


10-697393 
6967J9 


ll 


22 


1046 


991 397 


42 


3o326i 


1089 


23 


295286 


1045 


991372 


43 


3o39i4 
304567 
3o52i8 


1087 


696086 


37 


24 


295913 
296539 


io43 


99 1 346 


43 


1086 


695433 


36 


25 


1042 


991321 


43 


1084 


694782 


35 


26 


297164 


1040 


991 295 


43 


3o586g 


io83 


6941 3 1 


34 


2 


297788 


1039 


991270 


43 


3o65i9 
307168 


1081 


693481 


33 


298412 


1037 


991244 


43 


1080 


692832 


32 


29 


299034 


io36 


991218 


43 


307816 
3o8463 


1078 


692184 


3i 


3o 


299655 


io34 


991 i 9 3 


43 


1077 


691537 


3o 


3i 


9-300276 


1032 


9.991167 


43 


9.309109 


1075 


10-690891 


3 


32 


300895 


io3i 


991141 


43 


309754 


1074 


690246 


33 


3oi5i4 


1029 


991 n5 


43 


3 1 0399 


1073 


689601 


3 


34 


302132 


1028 


991090 


43 


3uo42 


1071 


688 9 58 
6883i 5 


35 


302748 


1026 


991064 


43 


3n685 


1070 


25 


36 


3o3364 


1025 


99io38 


43 


312327 
312968 


1068 


687673 


24 


37 


3o3o79 
304693 


1023 


99 1 1 2 


43 


1067 


687032 


23 


38 


1022 


990986 


43 


3i36o8 


io65 


6863 9 2 
685753 


22 


39 


306207 


1020 


990960 


43 


3i4247 


1064 


21 


40 


3o58i9 


IOI9 


990934 


44 


3 1 4885 


1062 


685u5 


20 


4i 


9>3o643o 


IOI7 
IOIO 


9.990908 


44 


9«3i5523 


1061 


10-684477 


\l 


42 


307041 


990882 


44 


3i6i5o 
316795 


1060 


683841 


43 


307650 
308259 


I0I4 


990855 


44 


io58 


6832o5 


\l 


44 


ioi3 


990829 
990803 


44 


3 n43o 


1057 


682570 


45 


308867 


ion 


44 


318064 


io55 


68i 9 36 
68i3o3 


i5 


46 


309474 


IOIC 


990777 


44 


318697 
3 19300 


io54 


14 


% 


3 1 0080 


1008 


990760 


44 


io53 


680670 


i3 


3io685 


1007 


990724 
990697 


44 


319961 


io5i 


68oo3o 
679408 


12 


49 


311289 
3n8 9 3 


ioo5 


44 


320692 


io5o 


11 


5o 


1004 


99067 . 


44 


321222 


1048 


678778 


10 


5i 


9 • 3 1 2495 


ioo3 


9.990645 


44 


9>32i85i 


1047 


10-678149 


I 


52 


3 1 3097 
3i36 9 8 


1001 


990618 


44 


322479 


io45 


677521 
676894 


53 


1000 


990591 
990565 


44 


323io6 


1044 


•7 


54 


314297 


998 


44 


323 7 33 
324358 


io43 


676267 


6 


55 


314897 


997 
996 


990538 


44 


1041 


675642 


5 


56 


3 1 5495 


9905 1 1 


45 


324983 


1040 


675017 
67439J 


4 


ll 


316092 
316689 


994 


990485 


45 


325607 


1039 


3 


993 


990458 


45 


32623i 


io37 


673769 


2 


5 9 


317284 


99 * 


990431 


45 


326853 


io36 


673147 
672525 


1 


60 


317879 


990 


990404 


45 


327475 


io35 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


f 


101 

















78° 



30 


LOGARITHMIC SINES, 


TANGENTS, ETC. Table IL 


12° 














w° 


i 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


t 
60 





tfiSB 


% 


9 • 990404 


45 


9-327475 


io35 


I0-672525 


i 


990378 


45 


328095 


io33 


671905 


& 


2 


319066 


III 


9903 5 1 


45 


328 7 i5 


1032 


671285 


3 


319658 


990324 


45 


329334 


io3o 


670666 


11 


4 


320249 


984 


990297 


45 


329953 


1029 


670047 


5 


320840 


9 83 


990270 


45 


33o370 


1028 


669430 
6688i3 


55 


6 


32i43o 


982 


990243 


45 


33ll8 7 


1026 


54 


I 


322019 


980 


9902l5 


45 


33i8o3 


1020 


668197 
66 7 5§2 
666967 
666354 


53 


322607 


979 


990188 


45 


332418 


1024 


52 


9 


323194 


977 


9901 6 1 


45 


333o33 


I023 


5i 


IO 


323780 


976 


990134 


45 


333646 


1021 


5o 


ii 


9-324366 


97 5 


9-990107 


46 


9-334259 


1020 


10-665741 


8 


12 


32495o 


973 


990079 


46 


334871 


IOI9 


665 1 29 


i3 


325534 


972 


990052 


46 


335482 


IOI7 


6645i8 


% 


14 


326117 


970 


990025 


46 


336093 


IOIO 


663907 
663 298 


i5 


326700 


^ 


989997 


46 


336702 


ioi5 


45 


16 


327281 


989970 


46 


3373i 1 


ioi3 


662689 


44 


17 


327862 . 


966 


989942 


46 


337919 


1012 


662081 


43 


18 


328442 


965 


989915 


46 


338527 


IOII 


661473 


42 


*9 


329021 


964 


989887 


46 


339i33 


IOIO 


660867 


41 


20 


329599 


962 


989860 


46 


339739 


1008 


660261 


4o 


21 


9-330176 


961 


9-989832 


46 


9-34o344 


1007 


10-659656 


ll 


22 


330753 


960 


989804 


46 


340948 
34i 552 


1006 


609002 


23 


33i329 


958 


989777 


46 


1004 


608448 


37 


24 


33i9o3 


9 5 7 


989749 


47 


342i55 


ioo3 


657845 


36 


25 


332478 


956 


989721 


47 


.342757 


1002 


657243 


35 


26 


333o5i 


954 


989693 


47 


343358 


1000 


656642 


34 


11 


333624 


9 53 


989665 


47 


343o58 


999 
998 


606042 


33 


334195 


952 


989637 


47 


344058 


655442 


32 


29 


334767 


95o 


989610 


47 


345i57 


997 


654843 


3i 


3o 


335337 


949 


989582 


47 


345755 


996 


654245 


3o 


3i 


9-335906 


948 


9-989553 


47 


9-346353 


994 


10-653647 


3 


32 


336475 


946 


989525 


47 


346949 


99 3 


653ooi 


33 


337043 


945 


989497 


47 


347040 


992 


652455 


2 


34 


337610 


944 


989469 


47 


348i4i 


991 


601859 


35 


338176 


943 


989441 


47 


. 348730 


990 


65i26o 


25 


36 


338742 


94i 


989413 


47 


349329 


988 


650671 


24 


37 


339307 


940 


9 8 9 385 


47 


349922 


987 


600078 


23 


38 


339871 


9 3 9 


989356 


47 


35ooi4 


986 


649486 
648894 


22 


39 


340434 


907 


989328 


47 


35no6 


9 85 


21 


40 


340996 


9 36 


989300 


47 


351697 


9 83 


6483o3 


20 


4i 


9-34i558 


9 35 


9-989271 


47 


9.352287 


982 


10-647713 


\l 


42 


342119 


934 


989243 


47 


352876 


981 


647124 


43 


342679 
34323g 


932 


989214 


47 


353465 


980 


646535 


\l 


44 


93i 


989186 


47 


354o53 


979 


640947 


45 


343797 


93o 


989157 
989128 


47 


354640 


977 


64o36o 


i5 


46 


344355 


929 


48 


355227 


976 


644773 


14 


% 


344912 


927 


989100 


48 


3558i3 


970 


644187 


i3 


345469 


926 


989071 


48 


356398 
356982 


974 


6436o2 


12 


49 


346024 


925 


989042 


48 


973 


643oi8 


11 


5o 


346579 


924 


989014 


48 


307066 


97i 


642434 


10 


5i 


9-347i34 


922 


9-988985 


48 


9-358i49 


970 


io-64i85i 


8 


52 


347687 


921 


988956 


48 


308731 


960 


641269 


53 


348240 


920 


988927 
988898 


48 


35 9 3i3 


9 6§ 


640687 


I 


54 


348792 


919 


48 


309893 


967 


6401 C7 
63 9 526 
638947 
638368 


55 


349343 


917 
916 


988869 


48 


36o474 


966 


5 


56 


349893 


988840 


48 


36io53 


965 


4 


u 


35o443 


9i5 


98881 1 


49 


36i632 


9 63 


3 


35 1 540 


9i4 


988782 


49 


362210 


962 


607790 


2 


59 


9i3 


988753 


49 


362787 


961 


6372i3 


1 


60 


352088 


911 


988724 


49 


363364 


960 


636636 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


102 















77° 



Table II. LOGARITHMIC SINES 


TANGENTS, ETC. 31 


13° 














160° 


/ 




Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 


9-352088 


911 


9-988724 


49 


9-363364 


960 


10-636636 


60 


i 


352635 


910 


988695 


49 


363940 
3645 1 5 


$1 


636o6o 


U 


2 


353i8i 


909 


988666 


49 


635485 


3 


353726 


908 


988636 


49 


365090 


9 5 7 


634910 
634336 


57 


4 


354271 


907 


988607 


49 


365664 


955 


56 


5 


3548i5 


9o5 


988578 


49 


366237 


9 ^ 


633763 


55 


6 


355358 


904 


988548 


49 


3668io 


9 53 


633190 


54 


7 


355901 


qo3 


988519 


49 


367382 


952 


6326i8 


53 


8 


356443 


902 


988489 


49 


367q53 


95 1 


632047 


52 


9 


356o84 
357624 


901 

899 


988460 


49 


368324 


g5o 


63 1476 


5i 


10 


988430 


49 


369094 


949 


630906 


5o 


ii 


9.358064 


898 


9 98840 1 


49 


9 • 369663 


948 


io-63o337 
629768 


% 


12 


3586o3 


897 


988371 


49 


370232 


946 


i3 


359141 


896 


988342 


49 


370799 


945 


629201 
628633 


47 


i4 


359678 


895 


988312 


5o 


371367 
371933 


944 


46 


i5 


36o2i5 


893 


988282 


5o 


943 


628067 


45 


16 


360752 


892 


988252 


5o 


372499 


942 


627501 


44 


\l 


361287 


891 


988223 


5o 


373064 


941 


626936 
626371 


43 


361822 


890 


988193 


5o 


373629 
37419J 
374706 


940 


42 


'9 


362356 


889 

888 


988163 


5o 


$ 


625807 


4i 


20 


362889 


988i33 


5o 


625244 


4o 


21 


9-363422 


887 


9-988103 


5o 


9.375319 


937 


10-624681 


39 


22 


363g54 


885 


988073 


5o 


375881 


9 35 


6241 19 
623558 


38 


23 


364485 


884 


988043 


5o 


376442 


934 


37 


24 


365oi6 


883 


9 88oi3 


5o 


377003 


9 33 


622997 
622437 
621878 


36 


25 


365546 


882 


987983 


5o 


377563 


932 


35 


26 


366075 


881 


987953 


5o 


378122 


9 3i 


34 


s 


3666o4 


880 


987922 
987892 


5o 


378681 


93o 


621319 


33 


367i3i 


879 


5o 


379239 


929 
928 


620761 


32 


29 


367659 
368 1 85 


877 


987862 


5o 


379797 


620203 


3i 


3o 


876 


987832 


5i 


38o354 


927 


619646 


3o 


3i 


9.368711 


8 7 5 


9.987801 


5i 


9.380910 


926 


10-619090 
6i8534 


2 


32 


369236 


874 


987771 


5i 


38i466 


925 


33 


369761 


8 7 3 


987740 


5i 


382020 


924 


617980 


27 


34 


370285 


872 


987710 


5i 


38 2 5 7 5 


923 


617425 


26 


35 


370808 


871 


987679 


5i 


383i29 


922 


616871 


25 


36 


37i33o 


870 


987649 
987618 


5i 


383682 


921 


6i63i8 


24 


37 


37i852 


869 


5i 


384234 


920 


61 5 7 66 


23 


38 


372373 


867 


987588 


5i 


384786 


919 


6i52i4 


22 


3 9 


372894 


866 


987557 
987526 


5i 


385337 
385888 


918 


6i4663 


21 


4o 


373414 


865 


5i 


9*7 


614112 


20 


4i 


9.373933 


864 


9.987496 


5i 


9-386438 


9i5 


io-6i3562 


IS 


42 


374452 


863 


987465 


5i 


. 386987 
38 7 536 


914 


6i3oi3 


43 


374970 


862 


987434 


5i 


9i3 


612464 


17 


44 


375487 


861 


987403 


52 


388o84 


912 


611916 


16 


45 


376003 


860 


987372 


52 


38863i 


911 


61 1J69 


i5 


46 


376519 
377035 


85 9 


987341 


52 


389178 


910 


610822 


14 


% 


858 


987310 


52 


389724 


900 
908 


610276 


i3 


377549 
378063 


85 7 
856 


987279 
987248 


52 


390270 


609730 


12 


49 


52 


390815 


907 


609185 


11 


5o 


378577 


854 


987217 


52 


391360 


906 


608640 


10 


5i 


9.379089 


853 


9-987186 


52 


9-391903 


905 


10-608097 
607533 


% 


52 


379601 
38ou3 


852 


987155 


52 


392447 


904 


53 


85 1 


987124 


52 


392989 


9o3 


60701 1 


7 


54 


380624 


85o 


987092 


52 


393631 


902 


6 $6469 


6 


55 


38u34 


849 

848 


987061 


52 


394073 


901 


605927 
6o5386 


5 


56 


38i643 


987030 


52 


394614 


900 
809 


4 


ll 


382i52 


847 

846 


986998 


52 


395 1 54 


604846 


3 


382661 


986967 


52 


395694 
396233 


898 


6o43o6 


2 


59 


383 1 68 


845 


986936 


52 


% 


603767 


1 


6o 


383675 


844 


986904 


02 


396771 


603229 




/ 


/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


:m 


5° 












76° 



32 


LOGARITHMIC SINES 


, TANGENTS, ETC. Table II 


14° 












165° 


t 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


t 


o 


9-383675 
384i82 


844 


9-986904 

986873 


52 


9.396771 


896 


10-603229 


6c 


i 


843 


53 


397309 


896 


602691 
602104 


is 


2 


384687 


842 


986841 


53 


397846 


8 9 5 


3 


385ig2 


841 


986809 
986778 


53 


3 9 8383 


894 


601617 


57 


4 


385697 


840 


53 


398919 
399456 


8 9 3 


601081 


56 


5 


386201 


83 9 
838 


986746 


53 


892 


6oo545 


55 


6 


386704 


9867U 


53 


399990 


891 


600010 


54 


I 


387207 


83 7 


986683 


53 


400024 


& 


599476 


53 


387709 


836 


986651 


53 


4oio58 


598942 


52 


9 


388210 


835 


986619 


53 


401591 


888 


598409 


5i 


10 


3887 1 1 


834 


986587 


53 


402124 


887 


597876 


5o 


ii 


9-389211 


833 


9-986555 


53 


9 -402656 


886 


10.597344 


% 


12 


3897 1 1 


832 


986523 


53 


403187 

403718 


885 


5 9 68i3 


i3 


390210 


83 1 


986491 
986459 


53 


884 


5g6282 


47 


U 


390708 


83o 


53 


404249 
404778 


883 


595701 


46 


i5 


391206 


828 


986427 
9863 9 5 


53 


882 


595222 


40 


16 


391703 


827 
826 


53 


4o53o8 


881 


594692 
5g4i64 


44 


n 


392199 


986363 


54 


4o5836 


880 


43 


18 


392695 


825 


98633i 


54 


4o6364 


ft 


593636 


42 


19 


393191 
3 9 3685 


824 


986299 


54 


406892 


593108 


41 


20 


823 


986266 


54 


407419 


877 


592581 


4o 


21 


9-394I79 
394673 


822 


9-986234 


54 


9.407945 


876 


io-592o55 


% 


22 


821 


986202 


54 


408471 


875 


591529 


23 


3 9 5i66 


820 


986169 


54 


408996 
409021 


874 


591004 


37 


24 


395658 


819 

818 


986137 


54 


874 


590479 


36 


25 


396150 


986 1 04 


54 


4ioo45 


8 7 3 


589950 


35 


26 


396641 


817 


986072 


54 


410569 


872 


58 9 43 1 


34 


2 


397132 


817 


986039 


54 


41 1092 


871 


588908 


33 


397621 


816 


986007 


54 


4u6i5 


870 


588385 


32 


29 


3981 1 1 


8i5 


985974 


54 


412137 
412658 


869 
868 


587863 


3i 


3o 


398600 


814 


985942 


54 


587342 


3o 


3i 


9-399088 


8i3 


9-985909 


55 


9-4i3i79 


867 


io-58682i 


3 


32 


399575 


812 


985876 


55 


413699 


866 


5863oi 


33 


400062 


811 


985843 


55 


414219 
4U738 


865 


585781 


27 


34 


400549 


810 


98581 1 


55 


864 


585262 


26 


35 


4oio3o 


809 


985778 


55 


4i5257 


864 


584743 


25 


36 


4oi52o 


808 


985745 


55 


4i5775 


863 


584225 


24 


u 


402005 


807 


985712 


55 


416293 


862 


583707 


23 


402489 


806 


985679 


55 


416810 


861 


583190 


22 


3 9 


402972 


8c5 


985646 


55 


417326 


860 


582674 


21 


40 


4o3455 


804 


9856i3 


55 


417842 


85g 


582158 


20 


4i 


9-4o3938 


8o3 


9-985580 


55 


9-4i8358 


858 


io-58i642 


\l 


42 


404420 


802 


985547 


55 


418873 


807 


581127 


43 


404901 


801 


9855i4 


55 


419387 


856 


58o6i3 


17 


44 


4o5382 


800 


985480 


55 


419901 


855 


580099 
579580 


16 


45 


4o5862 


$ 


985447 


55 


42041 5 


855 


i5 


46 


4o034i 


985414 


56 


420927 


854 


570073 
578560 


14 


47 


406820 


797 


98538i 


56 


421440 


853 


i3 


48 


407299 


796 


985347 


56 


421952 


852 


57S048 


12 


49 


407777 


7 9 5 


9853i4 


56 


422463 


85i 


577537 


11 


5o 


4o8254 


794 


985280 


56 


422974 


800 


577026 


10 


5i 


9-408731 


794 


9 985247 


56 


9-423484 


849 


io-5765i6 


I 


52 


409207 


7 9 3 


9852i3 


56 


423993 


848 


576007 


53 


409682 


792 


985i8o 


56 


424003 


848 


570407 
574989 


I 


54 


410157 


791 


985146 


56 


425oil 


847 


55 


4io632 


790 


985n3 


56 


4255i9 


846 


574481 


5 


56 


411106 


780 
788 


980079 
985o45 


56 


426027 


845 


573973 


4 


u 


411579 

412052 


56 


426534 


844 


573466 


3 


a 


985oii 


56 


427041 


843 


572959 
572403 


2 


59 


412524 


984978 


56 


427547 


843 


1 


60 


412996 


785 


984944 


56 


428002 


842 


571948 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotajig. 


D. 


Tang. 


1 


104 















75° 



Table II. LOGARITHMIC SINES 


TANGENTS, ETC. 33 


15° 














164° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


1 


o 


9.412996 


785 


9.984944 


57 


9-428052 


842 


10-571948 


60 


i 


413467 
4i3 9 38 


784 


9849IO 


57 


428558 


841 


571442 


u 


2 


7 S 


984876 


i 1 


429062 


840 


570938 


3 


4i44o8 


783 


984842 


57 


429566 


83 9 


570434 


n 


4 


414878 


782 


984808 


I 1 


430070 


838 


569930 


5 


4i5347 


781 


984774 


57 


43o573 


838 


56o427 


55 


6 


4i58i5 


780 


984740 


57 


431075 


83 7 


568925 


54 


I 


416283 


# 


984706 


57 


43i577 


836 


568423 


53 


416751 


984672 


^ 7 


432079 


835 


567921 


52 


9 


417217 


777 


984638 


57 


43258o 


834 


567420 


5i 


10 


417684 


776 


984603 


57 


433o8o 


833 


566920 


5o 


ii 


9-4i8i5o 


775 


9.984569 

984535 


57 


9-43358o 


832 


10-566420 


3 


12 


4i86i5 


774 


57 


434o8o 


832 


565920 


i3 


4i9°79 


773 


984500 


i 1 


43457Q 
435078 


83 1 


565421 


47 


U 


419544 


77 3 


984466 


5 7 


83o 


564922 


46 


i5 


420007 


772 


984432 


58 


435676 


829 


564424 


45 


16 


420470 


771 


984397 
984363 


58 


436073 


828 


563927 


44 


17 


420933 
42i3o5 
421857 


770 


58 


436570 


828 


56343o 


43 


18 


760 
768 


984328 


58 


437067 


827 


562933 


42 


19 


984294 


58 


437563 
438o59 


826 


562437 


4i 


20 


4223i8 


767 


984209 


58 


825 


561941 


4o 


21 


9.422778 


767 


9.984224 


58 


9-438554 


824 


io-56i446 


39 


22 


423238 


766 


984190 


58 


439048 


823 


560952 


38 


23 


423697 
4241 56 


7 65 


984i55 


58 


439543 


823 


56o457 


37 


24 


764 


984120 


58 


44oo36 


822 


559964 


36 


25 


4246i5 


7 63 


984085 


58 


44o529 


821 


559471 
558978 


35 


26 


425073 


762 


984050 


58 


441022 


820 


34 


11 


42553o 


761 


984015 


58 


44i5i4 


819 


558486 


33 


425987 
426443 


760 


983981 


58 


442006 


819 

818 


557994 
55 7 5o3 


32 


29 


760 


9 83 9 46 


58 


4424Q7 
442988 


3i 


3o 


426899 


7 5 9 


98391 1 


58 


817 


557012 


3o 


3i 


9.427354 


7 58 


9-983875 


58 


9.443479 
443968 


816 


10-556521 


s 


32 


427809 
42826J 


757 


9 8384o 


59 


816 


556o32 


33 


756 


9 838o5 


5 9 


444458 


8i5 


555542 


27 


34 


428717 


755 


983770 


£9 


444947 


814 


555o53 


26 


35 


429170 


754 


983735 


5 9 


445435 


8i3 


554565 


25 


36 


429623 


753 


983700 


59 


445923 


812 


554077 


24 


37 


430073 


752 


983664 


59 


44641 1 


812 


553589 


23 


38 


43o527 


752 


983629 


09 


446898 


811 


553 1 02 


22 


3 9 


430978 


7 5l 


983594 
9 83558 


59 


447384 


810 


5526i6 


21 


40 


431429 


75o 


59 


447870 


809 


552i3o 


20 


41 


9-431879 


749 


9.983523 


5 9 


9-448356 


809 

808 


io-55i644 


\l 


42 


432329 
432778 


740 

748 


983487 


59 


448841 


55n59 


43 


983452 


59 


449326 


807 


55o674 


17 


44 


433226 


746 


983416 


59 


449810 


806 


550190 


16 


45 


433675 


98338i 


5 9 


450294 


806 


549706 


i5 


46 


434122 


745 


983345 


59 


450777 


8o5 


54Q223 

548740 


14 


% 


434569 
435oi6 


744 


983309 
983273 
983238 


5 9 


45i26o 


804 


i3 


744 


60 


45i743 


8o3 


548257 


12 


49 


- 435462 


743 


60 


432225 


802 


547775 


11 


5o 


435908 


742 


983202 


60 


452706 


802 


547294 


10 


5i 


9-436353 


74i 


9.983166 


60 


9.453l87 


801 


IQ- 5468l3 


§ 


52 


436798 


740 


983 1 3o 


60 


453668 


800 


546332 


53 


437242 


74o 


983094 
983o58 


60 


454148 


799 


545852 


7 


54 


437686 


$ 


60 


454628 


799 


545372 


6 


55 


438129 


983022 


60 


455io7 


798 


544893 


5 


56 


438572 


]ll 


982986 


60 


455586 


19 1 
796 


5444U 


4 


ll 


439014 


982950 


60 


456o64 


543936 


3 


439456 


736 


982914 
982878 


60 


456542 


796 


543458 


2 


59 


439897 
440338 


735 


60 


457019 


79 5 


542981 
5425o4 


1 


6a 


734 


982842 


60 


457496 


794 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


J Of 


>° 














740- 



39 



34 


LOGARITHMIC SINES 


, TANGENTS, ETC. Table IL 


16 


3 












163° 


/ 


Sine. 


1 D 


Cosine. 


T). 


Tang. 


D. 


Cotang. 


/ 





9-44o338 


7 34 


9-982842 


60 


9-457496 


794 


io-5425o4 


60 


i 


440778 


733 


9828o5 


60 


457973 


793 


542027 


n 


2 


441218 


7 32 


982769 


61 


458449 


793 


54i55i 


3 


44 1 658 


7 3i 


982733 


61 


458925 


792 


541075 


57 


4 


442096 
442535 


7 3i 


9S2696 


61 


409400 


791 


540600 


00 


5 


73o 


982660 


61 


459875 


790 


540125 


55 


6 


442973 


3 


982624 


61 


46o349 
46o823 


W 9 


539651 


54 


I 


4434io 


982587 


61 


539177 


53 


443847 


727 


98255l 


61 


461297 


788 


538703 


52 


9 


444284 


727 


982514 


61 


461770 


788 


53823o 


5i 


10 


444720 


726 


982477 


61 


462242 


787 


537758 


5o 


ii 


9-445i55 


7 25 


9-98244I 


61 


9-462715 


786 


10.537285 


3 


12 


445590 


724 


982404 


61 


463 i 86 


785 


5368i4 


i3 


446025 


7 i3 


98236- 


61 


463658 


785 


536342 


47 


14 


446459 
446893 


7 23 


982331 


61 


464128 


784 


535872 


46 


id 


722 


982294 


61 


464599 


783 


53o4oi 


45 


16 


447326 


721 


982257 


61 


465o6g 


7 83 


53493i 


44 


n 


447759 


720 


982220 


62 


46553a 
466008 


782 


534461 


43 


18 


448191 


720 


982183 


62 


781 


533992 
533523 


42 


19 


448623 


719 


982146 


62 


466477 
466945 


780 


41 


20 


449004 


718 


982109 


62 


780 


533oo5 


40 


21 


9-449485 


7n 


9-982072 


62 


9.467413 


]]l 


10-532587 


It 


22 


4499i5 


716 


982035 


62 


467880 


532120 


23 


45o345 


716 


981998 


62 


468347 


778 


53i653 


37 


24 


400775 


7 i5 


981961 


62 


468814 


in 


53 1 1 86 


36 


25 


401204 


7i4 


981924 
981886 


62 


469280 


776 


530720 


35 


26 


45i63a 


7.3 


62 


469746 


775 


53o204 


34 


3 


452060 


7 i3 


981849 


62 


4702 1 1 


775 


529789 


33 


402488 


712 


981812 


62 


470676 


774 


529324 


32 


29 


452915 


711 


981774 


62 


47"4i 


773 


528859 


3i 


3o 


453342 


710 


981737 


62 


471605 


773 


5283 9 o 


3o 


3i 


9-453768 


710 


9-981700 


63 


9.472069 


772 


10027931 


29 


32 


454194 


700 
708 


981662 


63 


472532 


77i 


527468 


28 


33 


454619 


981625 


63 


472990 


77i 


527005 


27 


34 


455o44 


707 


981587 


63 


473407 


770 


526543 


26 


35 


455469 
455893 


707 


981049 


63 


473919 


769 


526081 


20 


36 


706 


98l5l2 


63 


47438i 


760 
768 


5256io 
520i58 


24 


37 


4563 1 6 


700 


9SI474 


63 


474842 


23 


38 


456739 


704 


9 8l436 


63 


4753o3 


767 


524697 
524237 


22 


3 9 


457162 


704 


981399 


63 


475763 


767 


21 


40 


457584 


703 


98l36l 


63 


476223 


766 


523 777 


20 


4i 


9- 458oo6 


702 


9-93l323 


63 


9.4766S3 


765 


io«5233i7 


\l 


42 


458427 
458848 


701 


981285 


63 


477U3 


7 65 


522858 


43 


701 


981247 


63 


477601 


764 


522399 


n 


44 


429268 


700 


981209 


63 


478009 


763 


521941 


16 


45 


459688 


690 


981171 


63 


4785i7 


763 


521483 


i5 


46 


460108 


698 


981 i33 


64 


478975 


762 


521025 


14 


47 


460527 


698 


981095 


64 


479432 


761 


520568 


i3 


48 


460946 


697 


■ 981007 


64 


4-9889 


761 


5201 1 1 


12 


49 


46 1 364 


696 


981019 


64 


48o340 


760 


519655 


11 


5o 


461782 


690 


9S0981 


64 


480801 


709 


519199 


10 


5i 


9-462199 


6 9 5 


9-980942 


64 


9-481257 


]U 


10.518743 


I 


52 


462616 


694 


9S0904 


64 


481712 


518288 


53 


463o32 


6 9 3 


980866 


64 


4S2167 


VI 


517833 


I 


54 


463448 


6 9 3 


980827 


64 


4S2621 


737 


517379 


55 


463864 


692 f 


980789 


64 


483075 


7 56 


516920 


5 


56 


464279 


691 


980750 


64 


483529 


700 


516471 


4 


57 


464694 


690 


980712 


64 


4839S2 


7 55 


5i6oi8 


3 


58 


465 1 08 


690 
689 
6S8 


980673 


64 


484435 


754 


5 1 5565 


2 


5 9 


465522 


9So635 


64 


4S48S7 


753 


5i5ii3 


1 


60 


465935 


980096 


64 


4S533g 


753 


5i466i 





r 


Cosine. 


D. | 


Sine. 


D. 


Cotang. 


D. 


Tang. 


106 















7S° 



Table II. LOGARITHMIC SINES 


TANGENTS. ETC. 


86 


11 c 














162° 


t 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 





9.465935 
466348 


688 


9. 9805^6 


64 


9.485339 


7 55 


io-5i466i 


60 


i 


688 


98o558 


64 


485791 


752 


514209 
5i3 7 58 


S3 


2 


466761 


687 


980519 


65 


486242 


7 5i 


3 


467173 


686 


980480 


65 


486693 


7 5i 


5i33o7 


U 


4 


467585 


685 


980442 


65 


487143 


75o 


5i2857 


5 


467996 


685 


98o4o3 


65 


487593 
488o43 


749 


512407 


55 


6 


468407 


684 


98o364 


65 


•749 


5iiq57 
5u5o8 


54 


7 


468817 


683 


98o325 


65 


488492 


748 


53 


8 


469227 


683 


980286 


65 


488941 


74^ 


5no59 


52 


9 


46963 j 


682 


980247 
980208 


65 


489390 

48 9 838 


747 


5io6io 


5i 


10 


470040 


681 


65 


746 


5ioi62 


5o 


ii 


9.470455 


680 


9-980169 


65 


9-490286 


746 


10.509714 


% 


12 


470863 


680 


98oi3o 


65 


490733 


745 


509267 


i3 


47I27I 


670 

678 


980091 
980062 


65 


491 180 


744 


508820 


47 


14 


471679 
472086 


65 


491627 
492073 


744 


5o8373 


46 


i5 


678 


980012 


65 


743 


507927 


45 


16 


472492 


677 


979973 


65 


492519 
492960 


743 


507481 


44 


\l 


472898 


676 


979934 
979895 
979855 


66 


742 


5o7o35 


43 


4733o4 


676 


66 


493410 


74i 


506590 


42 


19 


473710 


6 7 5 


66 


493854 


74o 


5o6i46 


41 


20 


4741 1 5 


674 


979816 


66 


494299 


74o 


505701 


4o 


21 


9.474519 
474Q2J 
475327 


674 


9.979776 


66 


9.494743 


74o 


io-5o5257 


is 


22 


673 


979737 


66 


495i86 


739 


5o48i4 


23 


672 


979608 


66 


49563o 


738 


504370 


37 


24 


47573o 


672 


66 


496073 


737 


503927 
5o3485 


36 


25 


476i33 


671 


979618 


66 


4965 1 5 


737 


35 


20 


476536 


670 


979579 


66 


496937 
-97399 


736 


5o3o43 


34 


11 


476938 
477340 


669 


979539 


66 


736 


502601 


33 


669 
668 


979499 
979459 


66 


497841 


735 


5o2i5o 
501718 


32 


29 


477741 


66 


498282 


734 


3i 


3o 


478142 


667 


979420 


66 


498722 


734 


501278 


3o 


3i 


9.478542 


667 


9.979380 


66 


9.499163 


7 33 


io-5oo837 


11 


32 


478942 
479342 


666 


979340 


66 


499603 


733 


5oo3o7 
499938 


33 


665 


979300 


67 


5ooo42 


7 32 


27 


34 


479741 


665 


979260 


67 


5oo48i 


7 3 1 


499519 


26 


35 


480140 


664 


979220 


67 


500920 


7 3i 


499080 
498641 


25 


36 


480539 


663 


979180 


67 


5oi359 


73o 


24 


3? 


480937 
481334 


663 


979U0 


67 


501797 


73o 


498203 


23 


38 


662 


979100 


67 


502235 


729 


4977^5 


22 


3 9 


481731 


661 


979 o5 9 


<? 


502672 


728 


497328 


21 


4o 


482128 


661 


979019 


67 


5o3i09 


728 


496891 


20 


4i 


9-482525 


660 


9.978979 


67 


9»5o3546 


727 


10.496454 


iq 


42 


482921 


65g 


978939 
978898 
978858 


67 


503982 


727 


496018 


18 


43 


4833i6 


65 9 


67 


5o44i8 


726 


495582 


17 


44 


483712 


658 


67 


5o4854 


7 25 


493146 


16 


45 


484107 


657 


978817 


67 


505289 


725 


4947 n 


i5 


46 


4845oi 


65 7 
656 


978777 


67 


5o5724 


724 


494276 


14 


47 


4848q5 


97873] 
978696 


6-7 


5o6i5o 
5o65g3 


724 


493841 


i3 


48 


485289 


655 


68 


7 23 


493407 
492973 
492040 


12 


49 


485682 


655 


978655 


68 


507027 


722 


11 


5o 


486075 


654 


97 86i5 


68 


507460 


722 


10 


5i 


9.486467 


653 


9-978574 


68 


9.507893 
5o8326 


721 


10.492107 


I 


52 


486860 


653 


978533 


68 


721 


491674 


53 


487251 


652 


9784q3 
978432 


68 


508759 


720 


491 241 


I 


54 


487643 
488o34 


65 1 


68 


509191 


719 


490809 


55 


65i 


97841 1 


68 


509622 


719 


490378 


5 


56 


488424 


65o 


978370 


68 


5ioo54 


718 


489946 


4 


ll 


488814 


65o 


978329 


68 


5io485 


718 


489315 


3 


489204 


649 
648 


978288 


68 


510916 
5n346 


717 


489084 


2 


5 9 


489593 
489982 


978247 


68 


716 


488654 


1 


60 


648 


978206 


68 


511776 


7i6 


488224 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


107 















'72° 



36 


LOGARITHMIC SINES 


TANGENTS, ETC. Table II 


18 


3 












16 1<> 


r 


Sine. 


D. 


Cosine. 


Id. 


Tang. 


D. 


Cotang. 


t 





9.489982 

490371 


648 


9-978206 


1 68 


9-511776 


716 


10-488224 


60 


i 


648 


978165 


68 


5l22o6 


716 


487794 


5 9 


2 


490759 


647 


978124 


68 


5i2635 


7 i5 


487365 


58 


3 


491 147 


646 


978083 


69 


5i3o64 


7i4 


486 9 36 
486507 


5 7 


4 


491535 


646 


978042 


69 


5 1 3493 


7U 


56 


5 


491922 
4923o8 


645 


978001 


69 


613921 
5 1 4349 


7 i3 


486079 


55 


6 


644 


977959 
977918 


69 


7*3 


48565i 


54 


I 


4926q5 


£44 


69 


514777 


712 


485223 


53 


493o8l 


643 


977877 
977835 


69 


5i52o4 


712 


484796 


52 


9 


493466 


642 


69 


5i563i 


711 


48436o 
483943 


5i 


10 


49385i 


642 


977794 


69 


5i6o57 


710 


5o 


ii 


9-494236 


641 


9-977752 


69 


9.516484 


710 


io.4835i6 


% 


12 


494621 


641 


9777H 


69 


016910 
317335 


709 


483090 


i3 


495oo5 


640 


977669 
977628 


69 


709 
708 


482665 


47 


14 


495388 


63 9 


69 


517761 


482239 


46 


i5 


493772 


63 9 


977086 


69 


618186 


708 


481814 


40 


16 


49604 


638 


977544 


70 


5i86io 


707 


481390 


44 


«7 


496537 


63 7 


9775o3 


70 


519034 


706 


480966 


43 


18 


496919 
497001 


637 


977461 


70 


619468 


706 


480342 


42 


'9 


636 


977419 


70 


519882 


7o5 


4801 18 


4i 


20 


497682 


636 


977377 


70 


52o3o5 


7o5 


479 6 9 5 


4o 


21 


9-498064 


635 


9.977335 


70 


9.520728 


7°4 


10.479272 


a 


22 


498444 


634 


977293 


70 


52ii5i 


703 


478849 


23 


498825 


634 


977231 


70 


521673 


7°3 


478427 


37 


24 


499204 


633 


977209 


70 


621993 


7o3 


478oo5 


36 


25 


499284 


632 


977167 
977125 


70 


522417 


702 


477583 


35 


26 


499963 
5oo342 


632 


70 


522838 


702 


477162 


34 


27 


63 1 


977083 


70 


623239 


701 


476741 


33 


28 


600721 


63 1 


977041 


70 


52368o 


701 


476320 


32 


29 


501099 


63o 


976999 


70 


524ioo 


700 


473900 


3i 


3o 


501476 


629 


970937 


70 


524520 


699 


476480 


3o 


3i 


9-5oi854 


629 
628 


9.976914 
976872 


70 


9.624940 
525339 
625778 


699 


10.475060 


3 


32 


50223l 


7i 


698 


474641 


33 


502607 


628 


97683o 


V 


698 


474222 


27 


,34 


502984 


627 


976787 


7i 


626197 


697 


4738o3 


26 


35 


5o336o 


626 


976745 


7i 


626616 


697 


473385 


25 


36 


5o3735 


626 


976702 


7i 


627033 


696 


472967 


24 


U 


5o4no 


626 


976660 


7i 


52745i 


696 


472049 


23 


5o4485 


625 


976617 


7i 


627868 


695 


472i32 


22 


3 9 


5o486o 


624 


976374 


71 


528286 


696 


471713 


21 


40 


5o5234 


623 


976532 


7i 


528702 


694 


471298 


20 


41 


9-5o56o8 


623 


9-976489 


7i 


9.529119 


693 


10-470881 


a 


42 


5o5q8i 
5o6354 


622 


976446 


7i 


629333 


6 9 3 


470465 


43 


622 


976404 


7i 


529951 
53o366 


6 9 3 


470049 


17 


44 


506727 


621 


976361 


7i 


692 


469634 


16 


45 


507099 


620 


976318 


7i 


530781 


691 


469219 
468804 


i5 


46 


607471 


620 


976275 


7i 


53 1 196 


691 


14 


47 


507843 


619 


976232 


72 


53i6u 


690 


468389 


i3 


48 


508214 


619 
618 


976189 
976146 


72 


532023 


690 


467073 


12 


49 


5o8585 


72 


532439 

532853 


689 


467361 


11 


5o 


D08956 


618 


976103 


72 


689 


467147 


10 


5i 


9 -509326 


617 


9-976060 


72 


9-533266 


688 


10-466734 


I 


52 


509696 


616 


976017 


72 


533679 


688 


466321 


53 


5ioo65 


616 


973974 


72 


534092 


687 


465908 


I 


54 


5io434 


6i5 


973930 
976887 


72 


5345o4 


687 


4654o6 
465o84 


55 


5io8o3 


6i5 


72 


534916 
535328 


686 


5 


56 


611172 


614 


97 5844 


72 


686 


464672 


4 


57 


5n54o 


6i3 


976800 


72 


535 7 3 9 


685 


464261 


3 


58 


511907 


6i3 


973767 


72 


536i5o 


685 


463S5o 


2 


5 9 


5i2275 


612 


975714 


72 


53656i 


684 


46343o 


1 


60 

/ 


5i2642 


612 


973670 


72 


536972 


6S4 


463023 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


108 















71° 



Table II. 


LOGARITHMIC SINES, TANGENTS, ETC. 3*7 


19 c 














160° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 


o 


9.512642 


612 


9-975670 


73 


9.536972 
537382 


684 


10-463028 


60 


i 


5 1 3009 
5i33 7 5 


611 


975627 


73 


683 


462618 


3 


2 


611 


97 5583 


73 


537792 
538202 


683 


462208 


3 


5i374i 


610 


975539 


73 


682 


461798 
46l389 


n 


4 


514107 


609 


975496 
975452 


73 


5386i 1 


682 


5 


5i4472 


609 
608 


73 


5390:0 


681 


460980 
460D71 


55 


6 


5i4837 


975408 


73 


539429 


681 


54 


I 


5-1 5202 


608 


975365 


73 


53 9 837 
540245 


680 


46oi63 


53 


5 i 5566 


607 


975321 


73 


680 


45 97 55 


52 


9 

IO 


5 1 5930 
516294 


607 
606 


975277 
975233 


73 


54o653 
54io6i 


679 
679 


459347 
458939 


5i 

5o 


ii 


9«5i6657 


6o5 


9.975189 
975i45 


73 


9-541468 


678 


10-458532 


i 


ia 


517020 


6o5 


73 


541875 
. 542281 


678 


458i25 


i3 


517382 


604 


975101 


73 


677 


457719 


47 


U 


517745 
518107 
5i8468 


604 


975o57 
975oi3 


73 


542688 


%l 


4573i2 


46 


i5 


6o3 


73 


543og4 


456906 
4565oi 


45 


16 


6o3 


974969 

974Q25 


74 


543499 
543oo5 
5443 10 


676 


44 


17 


. 518829 


602 


74 


675 


456095 


43 


18 


519190 
5i 9 55i 


601 


974080 


74 


675 


455690 

455285 


42 


19 


601 


974836 


74 


5447i5 


674 


41 


20 


519911 


600 


974792 


74 


545 1 19 


674 


454881 


40 


21 


9*520271 


600 


9.974748 


74 


9.545524 


6 7 3 


10-454476 


s 


22 


52o63i 


599 


9747o3 


74 


545928 
54633i 


6 7 3 


454072 
453669 
453265 


23 


520990 
52i349 


p? 


974659 


74 


672 


iz 


24 


974614 


74 


546735 


672 


25 


521707 
522066 


5 9 8 


974570 


74 


547i38 


671 


452862 


35 


26 


5 97 


974525 


74 


547540 


671 


45246o 


34 


3 


522424 


5 9 6 


97-4481 


74 


547943 
548345 


670 


452o57 


33 


522781 


5 9 6 


974436 


74 


670 


45i655 


32 


29 


5a3i38 


5 9 5 


974391 


74 


548747 


669 


45i253 


3i 


3o 


5234g5 


595 


974347 


75 


549149 


669 


45o85i 


3o 


3i 


9-523852 


594 


9-9743o2 


75 


9.549550 


668 


io-45o45o 


3 


32 


524208 


5 9 4 


974257 


75 


549951 


668 


45oo4o 


33 


524564 


K 


974212 


7 5 


55o352 


667 


449648 


27 


34 


524920 


593 


974167 


7 5 


55o752 


667 


449248 


26 


35 


525275 


592 


974122 


7 5 


55u53 


666 


448847 


25 


36 


52563o 


591 


974077 


75 


55i552 


666 


448448 


24 


ll 


5 2 5o84 
52633o 
526693 


591 


974o32 


75 


5519D2 
552351 


665 


448048 


23 


590 


973987 


7 5 


665 


447649 


22 


3 9 


590 


973942 
973897 


75 


55275o 


665 


44725o 


21 


4o 


527046 


589 


75 


553i49 


664 


44685i 


20 


4i 


9-527400 


589 


9. 973852 


7 5 


9.553548 


664 


10-446452 


\l 


42 


527753 
528io5 


588 


973807 


75 


553o46 
554344 


663 


446034 


43 


588 


973761 


75 


663 


445656 


17 


44 


528458 


S 7 


973716 


76 


554741 


662 


445259 


16 


45 


528810 


587 


973671 


76 


555i3 9 


662 


44486i 


i5 


46 


529161 


586 


973625 


76 


555536 


66 1 


444464 


14 


% 


52g5i3 


586 


97358o 


76 


555o33 
556329 
556 7 25 


661 


444067 


i3 


529864 


585 


973535 


76 


660 


443671 


12 


49 


53o2i5 


585 


973489 


76 


660 


443275 


11 


5o 


53o565 


584 


973444 


76 


557121 


65 9 


442879 


10 


5i 


9-53o9i5 


584 


9.973398 
973352 


76 


9 .55 7 5i7 
557913 
5583o8 


65 9 


10.442483 


I 


5a 


53i265 


583 


76 


65 9 


442087 


53 


53i6i4 


58z 


973307 


76 


658 


441692 


I 


54 


53 1 o63 

532JI2 


582 


973261 


76 


558 7 o3 


658 


441297 
44ooo3 
440009 
4401 1 5 


55 


58 1 


9732i5 


76 


559097 


65 7 


5 


56 


53266i 


58 1 


973169 


76 


559491 

55 9 885 


657 


4 


57 


533009 


58o 


973124 


76 


656 


3 


58 


533357 


58o 


973078 


76 


560270 
560673 


656 


439721 
439327 
438934 


2 


5 9 


533704 


579 

578 


973o32 


77 


655 


1 


60 


534o5a 


972986 


77 


56io66 


655 



/ 


/ 


Cosine. 


D. 


Sine. 


D. 


Co tang. 


D. 


Tang. 


10< 


\° 












10° 



88 


LOGARITHMIC SINES 


TANGENTS, ETC. Table II. 


20° 












159° 


r 




Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


1 ' 


9-534o5a 


5 7 8 


9.972986 


77 


9«56lo66 


655 


10.438934 


60 


i 


534399 
534745 


577 


972940 
972894 


77 


56l459 


654 


438541 


3 


2 


577 


77 


56i85i 


654 


438149 


3 


535092 
535438 


577 


972848 


77 


562244 


653 


437756 


57 


4 


576 


972802 


77 


562636 


653 


437364 


56 


5 


535 7 83 


5 7 6 


972755 


77 


563o28 


653 


436972 


55 


6 


536i2g 


5 7 5 


972709 
972663 


77 


563419 


652 


43658i 


54 


7 


536474 


5 7 4 


77 


5638n 


652 


436189 
43o 79 3 


53 


8 


5368i8 


574 


972617 


77 


564202 


65i 


52 


9 


537i63 


5 7 3 


972070 


77 


564593 


65i 


4304O7 


5i 


10 


537007 


573 


972524 


77 


5649»3 


600 


435oi7 


5o 


ii 


9.537851 


572 


9.972478 


]l 


9-565373 


65o 


10-434627 


3 


12 


538194 

538538 


572 


972431 


565763 


649 


434237 


i3 


5 7 i 


972385 


78 


566i53 


649 


433847 
433458 


% 


U 


538880 


5 7 i 


972338 


78 


566542 


649 
648 


i5 


539223 


570 


972291 


78 


566o32 

567J20 


433o68 


45 


16 


53g565 


5 7 o 


972245 


78 


648 


43268o 


44 


\l 


539907 


K 9 


972198 


78 


567709 
568098 
568486 


647 


432291 


43 


540249 


£8 


972i5i 


78 


647 


431902 


42 


'9 


540590 
540931 


568 


972io5 


78 


646 


43i5i4 


4i 


20 


568 


972o58 


78 


56&8 7 3 


646 


431127 


4o 


21 


9.541272 


56 7 


9-972011 


78 


9.569261 


645 


10-430739 


\% 


22 


54i6i3 


567 


97i9°4 


78 


569648 


645 


43o352 


23 


541953 


566 


971917 


78 


57oo35 


645 


429965 


37 


24 


542293 


566 


971870 


78 


570422 


644 


429578 


36 


25 


542632 


565 


971823 


78 


570809 


644 


429191 


35 


26 


542971 
5433io 


565 


971776 


78 


571 19D 


643 


4288o5 


34 


2 


564 


971729 


79 


571681 


643 


428419 
428o33 


33 


543649 


564 


971682 


79 


571967 


U2 


32 


29 


543987 
544325 


563 


97i635 


79 


572352 


642 


427648 


3i 


3o 


563 


971588 


79 


572738 


642 


427262 


3o 


3i 


9-544663 


562 


9-971540 


79 


9-573i23 


641 


10-426877 
426493 


3 


32 


545ooo 


562 


97U93 


79 


573507 


641 


33 


545338 


56 1 


971446 


79 


573892 


640 


426108 


27 


34 


545674 


56i 


971398 


79 


574276 


640 


425724 


26 


35 


54601 1 


56o 


97i3oi 


79 


574660 


63 9 


425340 


25 


36 


546347 


56o 


97i3o3 


79 


570044 


63g 


424956 


24 


37 


546683 


559 


971256 


79 


575427 


63 9 


424073 


23 


38 


547019 


559 


971208 


79 


575810 


638 


424190 


22 


3 9 


547354 


558 


971161 


79 


576193 


638 


423807 


21 


40 


547689 


558 


971 1 13 


79 


576576 


637 


423424 


20 


4i 


9.548024 


557 


9.971066 


80 


9.576959 
577J41 


63 7 


io-423o4i 


3 


42 


54835 9 
54869J 


- 557 
556 


971018 


80 


636 


422659 


43 


970970 


80 


577723 


636 


422277 
421896 


n 


44 


549027 


556 


970922 


80 


578104 


636 


16 


45 


549360 


555 


970874 


80 


. 578486 


635 


42i5i4 


i5 


46 


549693 


555 


970827 


80 


578867 
579248 


635 


42ii33 


14 


8 


55oo26 


554 


970779 


So 


634 


420752 


i3 


55o359 


554 


970731 


80 


579629 


634 


420371 


12 


49 


550692 


553 


970683 


80 


580009 


634 


419991 


11 


5o 


55io24 


553 


970635 


80 


58o389 


633 


419611 


10 


5i 


9-55i356 


552 


9-970586 


80 


9-580769 


633 


10-419231 

418801 


% 


52 


55i68-7 
552oi8 


552 


97o538 


80 


58n4o 
58i528 


632 


53 


552 


970490 


80 


632 


418472 


I 


54 


552349 


55i 


970442 


80 


581907 


632 


418093 


55 


552680 


55i 


970394 


80 


582286 


63 1 


4177U 


5 


56 


553oio 


55o 


97o345 


81 


582665 


63 1 


4n335 


4 


n 


553341 


55o 


970297 


81 


583o44 


63o 


416956 


3 


553670 


549 


970249 


81 


583422 


63o 


416078 


2 


5 9 


554ooo 


549 
548 


970200 


81 


5838oo 


629 


4:6200 


1 


60 


554329 


970152 


81 


584177 


629 


4i5823 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


IK 















r 


>9° J 



Table IL LOGARITHMIC SINES 


TANGENTS, ETC. 39 


21° 














158° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 





9.554329 
554658 


548 


9.970152 


8i 


9-584H7 
584555 


629 


io«4i5823 


60 


i 


548 


970103 


81 


629 
628 


4i5445 


% 


2 


554987 


547 


970055 


81 


584932 
5853o 9 


4i5o68 


3 


5553i5 


547 
546 


970006 


81 


628 


414691 


sz 


4 


555643 


969957 


81 


585686 


627 


4U3i4 


5 


555971 


546 


969909 


81 


586o62 


627 


4i3 9 38 


55 


6 


556299 


545 


969860 


81 


586439 


627 


4i356i 


54 


I 


556626 


545 


96981 1 


81 


5868 1 5 


626 


4i3i85 


53 


556953 


544 


969762 


81 


587190 


626 


412810 


52 


9 


557280 


544 


9697U 


81 


587566 


625 


412434 


5i 


10 


557606 


543 


969665 


81 


587941 


623 


412059 


5o 


ii 


9-557932 


543 


9-969616 


82 


9 .5883i6 


625 


io«4n684 


3 


12 


558258 


543 


969567 


82 


5886 9 i 


624 


4n3o9 


i3 


558583 


542 


969518 


82 


589066 


624 


4ioo34 
410060 


% 


14 


558909 


542 


969469 


82 


589440 


623 


i5 


559234 


54i 


969420 


82 


589814 


623 


410186 


45 


16 


55 9 558 


54i 


969370 


82 


590188 


b23 


409812 


44 


I? 


559883 


540 


969321 


82 


590562 


622 


409438 


43 


560207 


540 


969272 


82 


590935 
591008 


622 


409065 


42 


!9 


56o53i 


53 9 


969223 


82 


622 


408692 


41 


20 


56o855 


53 9 


969173 


82 


5 9 i68i 


621 


408319 


40 


21 


9-561178 


538 


9.969124 


82 


9.592054 


621 


10-407946 


ll 


22 


56i5oi 


538 


969075 


82 


592426 


620 


407374 


23 


56i824 


537 


969025 

968976 


82 


592799 


620 


407201 


37 


24 


562146 ' 


537 


82 


5 9 3i7i 


619 


406829 


36 


25 


562468 


536 


968926 
968877 


83 


593542 


619 


406438 


35 


26 


562790 


536 


83 


593914 


618 


406086 


34 


3 


563 1 1 2 


536 


968827 


83 


5 9 4285 


618 


4o57i5 


33 


563433 


535 


968777 
968728 


83 


5g4656 


618 


4o5344 


32 


29 


563 7 55 


535 


83 


59D027 
5 9 53g8 


617 


404973 


3i 


3o 


564075 


534 


968678 


83 


617 


404602 


3o 


3i 


9.564396 


534 


9.968628 


83 


9.595768 


617 


io-4o4232 


3 


32 


564716 


533 


968578 


83 


5 9 6i38 


616 


4o3862 


33 


565o36 


533 


968528 


83 


596508 


616 


403492 


27 


34 


565356 


532 


968479 


83 


596878 


616 


4o3i22 


26 


35 


565676 


532 


968429 


83 


597247 
597616 


6i5 


402753 


25 


36 


565995 
5663 14 


53 1 


968379 


83 


6i5 


402384 


24 


12 


53 1 


96832Q 
968278 


83 


597985 
5q8354 


6i5 


402015 


23 


566632 


53 1 


83 


614 


401646 


22 


39 


566g5i 


53o 


968228 


84 


598722 


614 


401278 


21 


4o 


567269 


53o 


968178 


84 


§99091 


6i3 


400909 


20 


4i 


9.567587 


529 


9-968128 


84 


9.599459 


6i3 


io-4oo54i 


\l 


42 


567904 

568222 


52 9 


968078 


84 


599827 


6i3 


400173 


43 


528 


968027 


84 


600194 


612 


399806 


H 


44 


. 56853g 


528 


967977 


84 


6oo5o2 


612 


399438 


16 


45 


568856 


528 


967027 


84 


600929 


611 


399071 


i5 


46 


569172 


527 


967876 


84 


601296 


611 


398704 


14 


4 I 


569488 


527 


967826 


84 


6oi663 


611 


3 9 8337 


i3 


4&i 


569804 


526 


967775 


84 


602029 
602396 


610 


397971 


12 


49 


570120 


526 


967725 


84 


610 


397605 


11 


5o 


570435 


525 


967674 


84 


602761 


610 


397239 


10 


5i 


9.570751 


525 


9.967624 


84 


9'6o3i27 


609 


10-396873 


i 


52 


071066 


524 


967573 


84 


6o3493 


609 


396507 


53 


57i38o 


524 


967522 


85 


6o3858 


609 


396142 


7 


54 


571695 


523 


96747I 


85 


604223 


608 


395777 


6 


55 


572009 
572323 


523 


96742I 


85 


6o4588 


608 


395412 


5 


56 


523 


967370 


85 


6o4953 


607 


395047 


4 


& 


572636 


j 522 


967319 
967268 


85 


6o53i7 


607 


394683 


3 


572900 


522 


85 


6o5682 


607 


394318 


2 


5 9 


573263 


521 


967217 


85 


606046 


606 


393954 


1 


6o 


573575 


521 


967166 


85 


606410 


606 


393690 




/ 


/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


11] 















68° 



40 


LOGARITHMIC SINES 


, TANGENTS, ETC. Table II. 


22 c 














157° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


1 


o 


Q-573575 


521 


9.967166 


85 


9 '606410 


606 


I 0^393590 


60 


1 


573888 


520 


9671 l5 


85 


606773 
607137 


606 


393227 
3 9 2863 


U 


2 


574200 


520 


967064 


85 


6o5 


3 


5745i2 


519 


967013 


85 


607500 


6o5 


3925oo 


5 7 


4 


574824 


519 


966961 


85 


607863 
6o8225 


604 


392137 

391775 


56 


5 


575i36 


5m 
5i8 


966910 
966859 
966808 


85 


604 


55 


6 


575447 
5 7 5 7 58 


85 


6o8588 


604 


39U12 


54 


7 


5i8 


85 


608950 
609312 


6o3 


39io5o 


53 


8 


576069 


5i 7 


966756 


86 


6o3 


3 9 o688 


52 


9 


576379 


5i 7 
5i6 


966705 


86 


609674 


6o3 


390326 
389964 


5i 


10 


576689 


966653 


86 


6ioo36 


602 


5o 


ii 


9.576999 

577309 
577618 


5i6 


9-966602 


86 


9-610397 


602 


IO-3896o3 


% 


12 


5i6 


96655o 


86 


610759 


H.2 


389241 
388880 


i3 


5i5 


966499 


86 


611120 


601 


47 


14 


577927 


5i5 


966447 
966395 


86 


61 1480 


601 


388520 


46 


i5 


5 7 8 2 36 


5i4 


86 


611841 


601 


388i59 


45 


16 


5 7 8545 


5i4 


966344 


80 


012201 


600 


?87799 
387439 


44 


12 


5 7 8853 


5i3 


966292 


86 


6i256i 


600 


43 


579162 


5i3 


966240 


86 


612921 


600 


387079 


42 


19 


579470 


5i3 


966188 


86 


613281 


599 


386719 


4i 


20 


579777 


5l2 


966l36 


86 


6i364i 


599 


38635 9 


4o 


21 


g.58oo85 


5l2 


9-966085 


87 


9-614000 


598 


10- 386ooo 


ll 


22 


58o3 9 2 


5n 


966o33 


87 


6i435o 
614718 


598 


385641 


23 


580699 
58ioo5 


5n 


965981 


87 


598 


385282 


ll 


24 


5u 


9 65 9 28 


87 


615077 
6i5435 


597 


384 9 23 


25 


58i3i2 


5io 


9 658i6 


87 


5 97 


384565 


35 


26 


58i6i8 


5io 


965824 


87 


615793 
6i6i5i 


597 


384207 


34 


S 


581924 


509 


965772 


87 


5 9 6 


383849 


33 


582229 
582535 


5og 


965720 


87 


616509 


5 9 6 


383491 


32 


29 


509 
5o8 


g65668 


87 


616867 


5 9 6 


383i33 


3i 


3o 


582840 


9656 1 5 


87 


617224 


5 9 5 


382776 


3o 


3i 


9. 583145 


5o8 


9-965563 


87 


9-617582 


5g5 


io-3824i8 


ll 


32 


583449 


507 


9655i 1 


87 


617939 


5 9 5 


382061 


33 


583754 


507 


965458 


87 


61829a 


594 


381705 


27 


34 


584o58 


5o6 


965406 


87 


6 1 8652 


5o4 


38i348 


26 


35 


584361 


5o6 


965353 


88 


619008 


594 


380992 


25 


36 


584665 


5o6 


9653oi 


88 


619364 


5 9 3 


38o636 


24 


37 


584968 


5o5 


965248 


88 


619720 


5 9 3 


380280 


23 


38 


585272 


5o5 


965195 


88 


620076 


5 9 3 


379924 


22 


3 9 


585574 


5o4 


965i43 


88 


620432 


592 


379068 


21 


4o 


585877 


5o4 


965090 


88 


620787 


592 


379213 


20 


4i 


Q- 586179 


5o3 


9-965037 


88 


9-621142 


592 


10-378858 


8 


42 


586482 


5o3 


964984 


88 


621497 


591 


3785o3 


43 


586783 


5o3 


96493 1 


88 


62i852 


591 


378148 


\l 


44 


587085 


502 


964879 


88 


622207 


590 


377793 
377439 


45 


587386 


502 


964826 


88 


622561 


590 


i5 


46 


587688 


5oi 


964773 


88 


622915 


590 


37708a 


14 


% 


587989 


5oi 


964720 


88 


623269 
623623 


58 9 


376731 


i3 


588289 


5oi 


964666 


89 


58 9 


. 376377 


12 


49 


5885 9 o 


5oo 


964613 


89 


623076 
62433o 


58 9 


376024 


11 


5o 


5888 9 o 


5oo 


964560 


89 


588 


370670 


10 


5i 


9.589190 


499 


9-964507 


89 


9-624683 


588 


10-375317 


I 


52 


589489 


499 


964454 


89 


625o36 


588 


374964 
374612 


53 


589789 
5 9 oo88 


499 


964400 


89 


625388 


58 7 


I 


54 


498 


964347 


89 


623741 


587 


374259 


55 


590387 


498 


964294 


89 


626093 


^ 7 


373907 


5 


56 


5 9 o686 


497 


964240 


89 


626445 


586 


373355 


4 


u 


590984 


497 


964187 
964133 


89 


626797 


586 


3732o3 


3 


591282 


4Q7 


89 


627149 


586 


372851 


2 


5 9 


5gi58o 


496 


964080 


89 


627501 


585 


372409 
372148 


1 


6o 


591878 


496 


964026 


89 


627852 


585 




t 


/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


112 















67° 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC 


41 


23° 














153° 


i 


Sine. 


D. 


Cosine. 


D. 


Tang. J 


D. 


Cotang. 


r 


o 


9.591878 


496 


9-964026 


89 


9 627852 
628203 


585 


lo«372i48 


60 


i 


592176 


495 


963972 


89 


585 


371797 

371440 


a 


2 


592473 


495 


9639J9 

9 6386d 


8 9 


628554 


585 


3 


592770 


495 


90 


028905 


584 


371095 


57 


4 


5g3o67 


494 


9 638ii 


90 


029255 


584 


370745 


56 


5 


5 9 3363 


494 


963757 


90 


629606 


583 


370394 


55 


6 


593659 
593955 


493 


963704 


90 


629956 
63o3o6 


583 


370044 


54 


I 


493 


96365o 


90 


583 


369694 


53 


594251 


493 


963596 


90 


63o656 


583 


369344 
368 99 5 


52 


9 


594547 


492 


963542 


90 


63ioo5 


582 


5i 


10 


594842 


492 


963488 


90 


63i355 


582 


368645 


5o 


ii 


9-595137 


491 


9-963434 


90 


9.631704 


582 


10-368296 


% 


12 


595432 


491 


963379 
963325 


90 


632o53 


58 1 


367947 
367698 
367200 


i3 


595727 


491 


90 


632402 


58.1 


% 


U 


596021 


490 


963271 


90 


63275o 


581 


i5 


5963i5 


490 
489 


9632 1.7 
963i63 


90 


633099 


58o 


366901 
366553 


45 


16 


596609 
59690J 


90 


633447 
633796 


58o 


44 


n 


489 


963108 


9' 


58o 


366205 


43 


18 


597196 


489 


963o54 


9> 


634i 43 


5 79 


365857 


42 
4i 
40 


l 9 


597490 
597783 


488 


962999 
962945 


9 1 


634490 
634838 


579 


3655io 


20 


488 


9i 


579 


365i62 


21 


9^98075 


487 


9-962890 


9 1 


o-635i85 


5 7 8 


io-3648i5 


u 


22 


5 9 8368 


487 


962836 


9 1 


635532 


5 7 8 


364468 


23 


5 9 866o 


487 


962781 


9* 


6358 7 9 


5 7 8 


364121 


37 


24 


598952 


486 


962727 


9 1 


636226 


5 77 


363774 


36 


25 


599244 


486 


962672 


9 1 


636572 


5 77 


363428 


35 


26 


599536 


486 


962617 


9 1 


636919 
637265 


577 


363o8i 


34 


3 


599827 


485 


962562 


9i 


577 


362735 


33 


6001 18 


485 


962508 


9 1 


63761 1 


5 7 6 


36238 9 


32 


29 


600409 


484 


962453 


9 1 


63 7 o56 
638302 


576 


362044 


3i 


3o 


600700 


484 


962398 


92 


5 7 6 


36i6 9 8 


3o 


3i 


9-600990 


484 


9-962343 


92 


9-638647 


5 7 5 


io-36i353 


2 


32 


601280 


483 


962288 


92 


638992 


5 7 5 


36ioo8 


33 


601570 


483 


962233 


92 


6393J7 


5 7 5 


36o663 


27 


34 


601860 


482 


962178 


92 


639682 


574 


36o3i8 


26 


35 


602 1 5o 


482 


962123 


92 


640027 


574 


359973 


25 


36 


6o243o 
602728 


482 


962067 


92 


640371 


574 


359629 


24 


37 


481 


962012 


92 


640716 


573 


359284 


23 


38 


603017 
6o33o5 


481 


961957 


92 


641060 


5 7 3 


358940 
3585 9 6 
358253 


22 


3 9 


481 


961902 


92 


641404 


5 7 3 


21 


40 


603594 


480 


961846 


92 


641747 


572 


20 


4i 


9-6o3882 


480 


9.961701 
961735 


92 


9-642091 
642434 


572 


10-357909 
357566 


\l 


42 


604170 
604457 
604745 


479 


02 


572 


43 


479 


961680 


92 


642777 


572 


357223 


17 


44 


479 
478 


961624 


9 3 


643i2o 


5 7 i 


356880 


16 


45 


6o5o32 


961569 
96i5i3 


93 


643463 


5 7 i 


356537 


i5 


46 


6o53i9 
6o56o6 


478 


93 


6438o6 


5 7 i 


356194 


14 


47 


478 


961458 


93 


644U8 


570 


355852 


i3 


48 


605892 


477 


961402 


93 


644490 


570 


3555io 


12 


49 


606179 
6o6465 


477 


961346 


9 3 


644832 


570 


355i68 


11 


5o 


476 


961290 


93 


645174 


56 9 


354826 


10 


5i 


9-606751 


476 


9-961235 


93 


9-6455i6 


56 9 


io- 354484 


i 


52 


607036 


476 


961 179 
961 1 23 


93 


645857 


56 9 


354U3 


53 


607322 


475 


93 


646199 


56g 
568 


3538oi 


7 


54 


607607 


475 


961067 


93 


646540 


35346o 


6 


55 


607892 


474 


961011 


9 3 


646881 


568 


353i 19 

352778 


5 


56 


608177 
608461 


474 


960955 


93 


647222 


568 


4 


u 


474 


960899 


9 3 


647562 


56 7 


352438 


3 


608745 


4 7 3 


960843 


94 


647903 


567 


352097 
3517D7 


2 


5 9 


609029 
6093 1 j 


473 


960786 


9 4 


648243 


567 


I 


60 


473 


960730 


9 4 


648583 


566 


35i4i7 





/ 


Cosine. 


D. 


Sine. 


D 


1 Cotang. 


D. 


Tang. 


/ 


11 


i° 












66° | 



42 


LOGARITHMIC SIXES, 


TANGENTS. ETC 


Table LL 


24° 














155° 


/ 


Sine. 


° 


Cosine. 


D. 


Tang. 


D. | 


Cotang. 


/ 





9-6o93i3 


4i3 


9*960730 


94 


9-648583 


566 


IO»35i4i7 


60 


i 


609597 
609880 


4?2 


960674 


94 


648923 


566 


35i077 


a 


2 


472 


960618 


94 


649263 


566 


35o737 


3 


610164 


472 


96o56l 


94 


649602 


566 


35o398 


57 


4 


610447 


47i 


96o5o5 


94 


649942 


565 


35oo58 


56 


5 


610729 


47i 


960448 


94 


65o28i 


565 


349719 


55 


6 


611012 


470 


960392 
96o335 


94 


65o620 


565 


34938o 


54 


7 


61 1 294 


470 


94 


650959 


564 


349041 


53 


8 


611576 


470 


960279 


94 


651297 


564 


348703 


52 


9 


6n858 


469 


960222 


94 


65i636 


564 


348364 


5i 


10 


612140 


469 


960163 


94 


651974 


563 


348026 


5o 


ii 


9-612421 


469 
468 


9-960109 


9 5 


9-652312 


563 


10-347688 


% 


12 


612702 


960052 


9 5 


65265o 


563 


34735o 


i3 


612983 


468 


939993 
939938 
959882 


9 5 


652 9 88 


563 


347012 


47 


U 


6i3264 


467 


9 5 


6533 ?6 


562 


346674 


46 


i5 


6i3545 


467 


9 5 


653663 


562 


346337 


45 


16 


6i382D 


467 


959823 


9 5 


654ooo 


562 


346000 


44 


17 


614100 


466 


959768 


9 5 


654337 


56i 


345663 


43 


18 


6i4385 


466 


95971 1 


9 5 


654674 


56 1 


345326 


42 


19 


6i4665 


466 


959654 


9 5 


655ou 


56 1 


344989 


41 


20 


614944 


465 


959596 


9 5 


655348 


56i 


344652 


4o 


21 


g-6i5223 


465 


9.959539 


9 5 


9-655684 


56o 


io-3443i6 


u 


22 


6i55o2 


465 


959482 


9 5 


656o2o 


56o 


343980 


23 


610781 


464 


959425 


9 5 


656356 


56o 


343644 


37 


24 


616060 


464 


959368 


9 5 


656692 


539 


3433o8 


36 


25 


6i6338 


464 


939310 


96 


657028 


559 


342972 


35 


26 


616616 


463 


959233 


96 


657364 


559 


342636 


34 


3 


616894 


463 


959195 
939138 


96 


657699 
658o34 


559 


3423oi 


33 


617172 


462 


96 


558 


341966 


32 


29 


617450 


462 


959080 


96 


658369 


558 


34i63i 


3i 


3o 


617727 


462 


959023 


96 


658704 


558 


341296 


3o 


3i 


9-618004 


461 


9-958965 


96 


9.659039 
659373 


558 


10-340961 


29 


32 


618281 


461 


958008 
95885o 


96 


557 


340027 


28 


33 


6i8558 


461 


96 


659708 


55-) 


340292 


27 


34 


6i8834 


460 


958792 
958734 


96 


660042 


537 


33 99 58 


26 


35 


61911c 


460 


66 


660376 


55 7 


339624 


25 


36 


6i 9 386 


460 


9 586 77 


96 


660710 


556 


339290 


24 


37 


619662 


459 


958619 


96 


66io43 


556 


33S 9 57 


23 


38 


619938 


45q 


95856i 


96 


661377 


556 


333623 


22 


39 


620213 


45c 

458 


g585o3 


97 


661710 


555 


338290 


21 


4o 


620488 


958445 


97 


662043 


555 


337937 


2C 


4i 


9-620763 


458 


9-953387 


97 


9-662376 


555 


10-337624 


3 


42 


62io38 


457 


958329 


97 


662709 


554 


337291 


43 


62i3i3 


45 7 


958271 


97 


663042 


554 


336 9 58 


! 7 


44 


621587 


45 7 


9582i3 


97 


663375 


554 


336625 


16 


45 


621861 


456 


93S134 


97 


663707 


554 


336293 


i5 


46 


622i35 


456 


938096 


97 


664039 


553 


335961 


14 


47 


622409 


456 


95So38 


97 


664371 


553 


335629 


i3 


48 


622682 


455 


957979 


97 


664703 


553 


335297 


12 


49 


622956 


455 


957921 


97 


665o35 


553 


334965 


11 


5o 


623229 


455 


957863 


97 


665366 


552 


334634 


10 


5i 


g-6235c^ 


454 


9-957804 


97 


9 -665698 


552 


io-3343o2 


3 


52 


623774 


454 


9 5 77 46 


98 


666029 


552 


333971 


53 


624047 


454 


9576S7 


98 


66636o 


55i 


333640 


- 


54 


624319 


453 


957628 


98 


666691 


55i 


3333o9 


6 


55 


624591 


453 


957570 


98 


667021 


55i 


332970 
332643 


5 


56 


624863 


453 


937311 


98 


667352 


55i 


4 


Si 


625i35 


452 


9^7452 


9S 


6676S2 


55o 


3323i8 


3 


625406 


452 


957393 
957335 


98 


66801 3 


55o 


3319S7 


2 


5 9 


625677 
625948 


45a 


98 


668343 


55o 


33i65"7 


1 


6o 


45i 


937276 


98 


668673 


55o 


33i327 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


IU 


t° 












65° 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC. 48 


25° 












154° 


/ 


1 Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


f 





9-625948 


45 1 


9.957276 


9? 


9-668673 


55o 


jo-33i327 
330998 


60 


i 


626219 


45i 


957217 
957158 


98 


669002 


549 


ll 


2 


626490 


45 1 


98 


669332 


549 


33o668 


3 


626760 


45o 


*9 5 7099 


9 S? 


669661 


549 


33o339 


5 7 


4 


627030 


45o 


9^7040 


98 


669991 
670320 


548 


330009 


56 


5 


627300 


45o 


9O6981 


98 


548 


329680 


55 


6 


627570 


449 


956Q2I 

956862 


99 


670649 


548 


329351 


54 


I 


627840 


449 


99 


670977 
67l3o6 


548 


329023 
328694 


53 


628109 
628378 


449 

448 


9568o3 


99 


547 


52 


9 


956744 


99 


671635 


547 


328365 


5i 


10 


628647 


448 


956684 


99 


671963 


547 


328037 


5o 


ii 


9-628916 


45 7 


9-956625 


99 


9-67229! 


547 


10-327709 


% 


12 


629185 


447 


956566 


99 


672619 


546 


327381 


i3 


629453 


447 
446 


9565o6 


99 


672947 


546 


327o53 


% 


i4 


629721 


956447 


99 


673274 


546 


326726 


i5 


629989 


446 


956387 


99 


673602 


546 


326398 


45 


16 


63o25"7 


446 


956327 


99 


673929 


545 


326071 


44 


\l 


63o524 


446 


956268 


99 


674257 


545 


325743 


43 


630792 
63 1009 


445 


956208 


100 


674584 


545 


3254i6 


42 


19 


445 


956148 


100 


6749 I I 


544 


325089 
324763 


4i 


20 


63i326 


445 


956089 


100 


675237 


544 


4o 


21 


9«63i593 


444 


9-956029 


100 


9-675564 


544 


10-324436 


3 


22 


63i85g 


444 


955969 


100 


675890 


544 


324110' 


23 


632120 


444 


955909 


100 


676217 


543 


323 7 83 


37 


24 


632392 


443 


955849 


100 


676543 


543 


323457 


36 


25 


632658 


443 


955789 


100 


676869 


543 


323i3r 


35 


26 


632923 


443 


955729 


100 


677194 


543 


322806 


34 


2 


633i8g 


442 


955669 


100 


677520 


542 


322480 


33 


633454 


442 


955609 
955548 


100 


677846 


542 


322i54 


32 


29 


633719 


442 


100 


678171 


542 


321829 


3i 


3o 


633984 


44i 


955488 


100 


678496 


542 


32i5o4 


3o 


3i 


9-634249 


44i 


9-955428 


101 


9-678821 


54i 


10-321179 


3 


32 


6345i4 


440 


9 55368 


101 


679146 


54i 


32o854 


33 


634778 


44o 


955307 


101 


679471 


54i 


320529 

320205 


27 


34 


635o42 


44o 


955247 


101 


679795 


54i 


26 


35 


6353o6 


439 


955i86 


101 


680120 


54o 


3i 9 88o 


25 


36 


635570 


439 


955126 


101 


68o444 


54o. 


319556 


24 


37 


635834 


43 9 


955o65 


101 


680768 


54o 


3lQ232 


23 


38 


636097 


438 


955oo5 


101 


681092 


54o 


318908 


22 


3 9 


63636o 


438 


954944 


101 


681416 


53 9 


3 1 8584 


21 


40 


636623 


438 


954883 


101 


681740 


53 9 


318260 


20 


4i 


9-636886 


437 


9-954823 


101 


9-682063 


53 9 


10-317937 
31761J 


\l 


42 


637148 


437 


954762 


101 


682387 


53 9 

538 


43 


63741 1 


437 


954701 


101 


682710 


317290 
316967 


\l 


44 


63 7 6 7 3 


437 


954640 


101 


683o33 


538 


45 


637935 
638i 9 7 


436 


954579 


101 


683356 


538 


3i6644 


i5 


46 


436 


9545i8 


102 


683679 


538 


3i632i 


14 


% 


638458 


436 


954457 


102 


684001 


53 7 


3 1 5999 


i3 


638720 


435 


904396 
954335 


102 


684324 


53 7 


3i56 7 6 


12 


49 


6.38981 


435 


102 


684646 


53 7 


3 1 5354 


11 


5o 


639242 


435 


904274 


102 


684968 


53 7 


3i5o32 


10 


5i 


9-6395o3 


434 


9-954213 


102 


9-685290 


536 


iO'3i47io 


8 


52 


639764 


434 


954152 


102 


6856ia 


536 


3 U388 


53 


640024 


434 


954090 


102 


685 9 34 


536 


3 1 4066 


7 


54 


640284 


433 


954029 
953968 


102 


686255 


536 


3.i3745 


6 


55 


64o544 


433 


102 


686577 
686898 


535 


3i3423 


5 


56 


640804 


433 


953906 


102 


535 


3i3io2 


4 


& 


641064 


432 


953845 


102 


687219 


535 


312781 


3 


641324 


432 


903783 


102 


687540 


535 


312460 


2 


59 


641 583 


432 


953722 


io3 


687861 


534 


3 1 2 1 3q 

3ii8i8 


I 


60 


641842 


43 1 


95366o 


io3 


688182 


534 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


115 















64° 



44 


LOGARITHMIC SINES, 


TANGENTS, ETC. Table IL 


26° 














158° 


t 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 





9-641842 


43 1 


9.953660 


io3 


9-688182 


534 


IO-3ll8l8 


60 


i 


642101 


43 1 


953599 


io3 


688502 


534 


3 1 U98 


u 


2 


642360 


43 1 


9 5353 7 


io3 


688823 


534 


311177 


3 


642618 


43o 


953475 


io3 


689143 


533 


3lo857 


57 


4 


642877 
643 i 35 


43o 


9534l3 


io3 


68 9 463 


533 


3io537 


56 


5 


43o 


953352 


io3 


689783 


533 


310217 


55 


6 


6433o3 
64365o 


43o 


953290 


io3 


690103 


533 


309897 


54 


7 8 


429 


953228 


lo3 


690423 


533 


309577 
3oo258 
3o8 9 38 


53 


643908 


429 


953l66 


,o3 


690742 


532 


52 


9 


644i65 


429 

428 


953104 


io3 


691062 


532 


5i 


IO 


644423 


953o42 


io3 


6 9 i38i 


532 


308619 


5o 


ii 


9.644680 


428 


9-952980 


104 


9.691700 


53 1 


io«3o83oo 


i 


12 


644936 


428 


952918 
952855 


104 


692019 
692338 


53 1 


307981 


i3 


645ig3 


427 


104 


53 1 


307662 


% 


14 


645430 


427 


952793 
95273l 


104 


692656 


53 1 


307344 


i5 


645706 


427 


104 


692975 


53 1 


307025 


45 


16 


645962 


426 


952669 


104 


693293 


53o 


306707 
3o6388 


44 


n 


646218 


426 


952606 


104 


693612 


53o 


43 


18 


646474 


426 


952544 


104 


693930 


53o 


306070 


42 


»9 


646729 


425 


952481 


104 


694248 


53o 


3o5752 


4i 


20 


646984 


425 


952419 


104 


694566 


529 


3o5434 


4o 


21 


9.647240 


425 


9 952356 


104 


9.694883 


529 


io-3o5ii7 


it 


22 


647494 


424 


952294 
95223i 


104 


695201 


629 


304799 
304482 


23 


647749 
648004 


424 


104 


6 9 55 1 8 


529 


37 


24 


424 


9 52i68 


io5 


6 9 5836 


529 


3o4i64 


36 


25 


648258 


424 


902106 


io5 


696153 


528 


3 03847 


35 


26 


6485i2 


423 


952043 


io5 


696470 


528 


3o353o 


34 


3 


648766 


423 


951980 


io5 


696787 


5 2 8 


3o32i3 


33 


649020 


423 


951917 
951834 


io5 


697103 


528 


302897 
3o258o 


32 


29 


649274 


422 


103 


697420 


527 


3i 


3o 


649527 


422 


9 5i 79 i 


io5 


697736 


527 


302264 


3o 


3i 


9.649781 


422 


9.951728 


io5 


9.698053 


52 7 


10-301947 


% 


32 


65oo34 


422 


9Di665 


io5 


6 9 836o 


527 


3oi63i 


33 


650287 


421 


951602 


io5 


698683 


526 


3oi3i5 


3 


34 


65o539 


421 


95i539 


io5 


699001 


526 


300999 


35 


650792 


421 


951476 


io5 


699316 


526 


3oo684 


23 


36 


65 1 o44 


420 


951412 


:o5 


699632 


526 


3oo368 


24 


37 


65 1 297 


420 


95i34g 


106 


699947 


526 


3ooo53 


23 


38 


65 1 549 


420 


951286 


106 


700263 


525 


299737 


22 


39 


65 1 800 


419 


951222 


106 


700578 


523 


299422 


21 


4o 


652o52 


419 


951169 


106 


700893 


525 


299107 


20 


4i 


9-6523o4 


419 

418 


9-951096 
95ioj2 


106 


9.701208 


524 


10-298792 


It 


42 


652555 


106 


7oi523 


524 


298477* 


43 


6528o6 


418 


950968 


106 


701837 


524 


2 9 8i63 


\l 


44 


653o57 


418 


95ooo5 


106 


702152 


524 


297848 


45 


6533o8 


418 


960841 


106 


702466 


024 


297534 


i5 


46 


653558 


417 


950778 


106 


702781 


5 2 3 


297219 


14 


% 


6538o8 


417 


950714 


106 


703095 


523 


296903 


i3 


654059 


416 


95o65o 


106 


703409 


5s3 


296391 


12 


49 


6543og 
654558 


95o586 


106 


703722 


5 2 3 


296278 


11 


5o 


416 


95o522 


107 


704036 


522 


293964 


10 


5i 


9.654808 


416 


9-95o458 


107 


9.704350 


522 


io-29565o 


I 


52 


655o58 


416 


9 5o3o4 
95o33o 


107 


704663 


522 


295337 


53 


6553o7 


4i5 


107 


704976 


522 


293024 


I 


54 


655556 


4i5 


950266 


107 


703290 


522 


294710 


55 


6558o5 


4i5 


960202 


107 


7o56o3 


521 


294307 
294084 


5 


56 


656o54 


4i4 


95oi38 


107 


705916 


521 


4 


57 


6563o2 


4U 


960074 


107 


706228 


521 


293772 


3 


58 


656551 


4i4 


930010 


107 


706541 


521 


293450 
293146 


t 


5 9 


656799 


4i3 


949945 
949881 


107 


706834 


521 


1 


6o 


657047 


4i3 


107 


707166 


520 


292834 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


/ 


lit 


>° 












68° 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC. 45 


27 c 














152° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


f 


o 


9-657047 
65729D 


4i3 


9-949881 


107 


0*707166 


520 


10.292834 


60 


i 


4i3 


949816 


107 


707478 


520 


292522 


it 


2 


657542 


412 


949752 


108 


707790 
708102 


520 


292210 


3 


657790 
658o3 7 


412 


949688 


520 


291898 
291586 


n 


4 


412 


949623 


108 


7084U 


519 


5 


658284 


412 


949558 


108 


708726 


519 


291274 


55 


6 


65853i 


411 


949494 


108 


709037 


519 


290963 


54 


I 


658778 


411 


949429 


108 


709349 


519 


290651 


53 


659025 


411 


949364 


108 


709660 


5ig 
5i8 


290340 


52 


9 


659271 


410 


949300 


108 


709971 
710282 


290029 

289718 


5i 


10 


659517 


410 


949235 


108 


5i8 


5o 


ii 


9-659763 


410 


9-949170 


108 


9-7I0593 


5i8 


10-289407 
289096 

288785 


8 


12 


660009 
66o25d 


409 


949105 


108 


710904 


5i8 


i3 


409 


949040 
948975 


108 


71 I2l5 


5i8 


47 


i4 


66o5oi 


409 


108 


7ii525 


5i 7 


288475 


46 


i5 


660746 


409 
408 


948910 

948845 


108 


7ii836 


5i 7 


288164 


45 


16 


660991 
66i236 


108 


712146 


5i 7 


287854 


44 


\l 


408 


. 948780 


109 


712456 


5i 7 


287544 


43 


661481 


408 


948715 


109 


712766 


5i6 


287234 
286924 


42 


'9 


661726 


407 


9486DO 


109 


713076 
713386 


5i6 


4i 


20 


661970 


407 


948584 


109 


5i6 


286614 


4o 


21 


9-662214 


407 


9.948519 


109 


9.713696 


5i6 


io«2863o4 


u 


22 


66245q 
66270J 


407 


948454 


109 


714005 


5i6 


285995 
285686 


23 


406 


948388 


109 


7i43i4 


5i5 


37 


24 


662946 


406 


948323 


109 


714624 


5i5 


285376 


36 


25 


663190 
663433 


406 


948257 


109 


714933 


5i5 


285067 
284758 


35 


26 


4o5 


948192 


109 


715242 


5i5 


34 


2 


663677 


4o5 


948126 


109 


7 1 555 1 


5i4 


284449 


33 


663920 


4o5 


948060 


109 


7i586o 


5i4 


284140 


32 


29 


664 1 63 


4o5 


94799 5 


no 


716168 


5i4 


283832 


3i 


3o 


664406 


404 


947929 


no 


716477 


5i4 


283523 


3o 


3i 


9-664648 


404 


9-947863 


no 


9 716785 


5i4 


io-2832i5 


3 


32 


664891 


404 


947797 
947665 


no 


717093 


5i3 


282907 


33 


665(33 


4o3 


no 


717401 


5i3 


282099 


27 


34 


665375 


4o3 


no 


717709 


5i3 


282291 


26 


35 


6656i7 


4o3 


947600 


no 


718017 
7i8325 


5i3 


281983 


25 


36 


66585 9 


402 


947533 


no 


5i3 


281675 


24 


U 


666100 


402 


947467 


no 


7i8633 


5l2 


281367 


23 


666342 


402 


947401 


no 


718940 


5l2 


281060 


22 


3 9 


666583 


402 


947335 


no 


719248 


5l2 


280752 


21 


4o 


666824 


4oi 


947269 


no 


719555 


5l2 


280445 


20 


4i 


9-667065 


4oi 


9.947203 


no 


9.719862 


5l2 


io-28oi38 


!8 


42 


6673o5 


401 


947i36 


m 


720169 


5n 


279831 


43 


667546 


4oi 


947070 


in 


720476 


5n 


279024 


17 


44 


667786 


400 


947004 


n 1 


720783 


5n 


279217 


16 


45 


668027 


400 


946937 
946871 


in 


721089 
721396 


5n 


27891 1 


i5 


46 


668267 
6685o6 


400 


in 


5n 


278604 


14 


3 


399 


946804 


in 


721702 


5io 


278298 


i3 


668746 


3 99 


946738 


in 


722009 
7223i5 


5io 


277991 


12 


49 


668986 


399 


946671 


in 


5io 


277685 


11 


5o 


669225 


3 99 


946604 


111 


722621 


5io 


277379 


10 


5i 


9-669464 


398 


9-946538 


in 


9.722927 


5io 


10.277073 


i 


52 


669703 


3q8 


946471 


in 


723232 


509 


276708 


53 


669942 


3 9 8 


946404 


in 


723538 


509 


276462 


I 


54 


670181 


397 


946337 


in 


723844 


509 


276106 


55 


670419 
6 7 o658 


397 


946270 


112 


724149 


5og 


27585i 


5 


56 


397 


946203 


112 


724454 


5o 9 


275546 


4 


u 


670896 
671134 


3 97 


946 1 36 


112 


724760 


5o8 


275240 


3 


3 9 6 


946069 


112 


725o65 


5o8 


274935 


2 


59 


671372 


3 9 6 


946002 


112 


725370 


5o8 


274630 


1 


60 


671609 


3 9 6 


945935 


112 


723674 


5o8 


274326 




/ 


/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


iti 















62° 



46 


LOGARITHMIC SIXES, 


TANGENTS, ETC 


Table IL 


28° 














151° 


/ 


Sine. 


D. 


Cosine. 


D. | 


Tang. 


D. | 


Cotang. 


f 





9.671609 


3 9 6 


9-945935 


112 


9.725674 


008 


I0.274326 


60 


i 


671847 


395 


945868 


112 


725979 


5o8 


274021 


SI 


2 


672084 


390 


9458oo 


112 


726284 


507 


273716 


3 


672321 


393 


945733 


112 


726588 


507 


273412 


57 


4 


672558 


3g5 


945666 


IJ2 


726892 


507 


273108 


56 


5 


672795 
673o32 


394 


945598 
94553i 


112 


727197 


507 


2728o3 


55 


6 


3 9 4 


112 


727501 


507 


272499 


54 


I 


673268 


394 


945464 


Il3 


727805 
728109 


5o6 


272195 


53 


673300 


3 9 4 


945396 


Il3 


5o6 


271891 


52 


9 


673741 


3 9 3 


945328 


1*3 


728412 


5o6 


271588 


Si 


10 


673977 


3 9.3 


945261 


n3 


728716 


006 


271284 


5o 


ii 


9-674213 


3 9 3 


9-945i93 


n3 


9.729020 


5o6 


10-270980 


4 2 


12 


674448 


392 


945i25 


n3 


729323 


5o5 


270677 


48 


i3 


674684 


392 


945o58 


n3 


729626 


5o5 


270374 


47 


i4 


6749IQ 
675l55 


392 


944990 


n3 


729929 
73o233 


5o5 


270071 


46 


i5 


392 


944922 
944804 


n3 


5o5 


269767 


45 


16 


675390 


3gi 


n3 


73o535 


5o5 


269465 


44 


n 


675624 


391 


944786 


n3 


7 3o838 


5o4 


269162 


43 


18 


675859 


391 


9447i8 


n3 


731141 


5o4 


26885g 


42 


l 9 


676094 


391 


94465o 


n3 


731444 


5o4 


268556 


4i 


20 


676328 


390 


944582 


114 


731746 


004 


268254 


4o 


21 


9-676562 


390 


9-9445U 


114 


9.732048 


5o4 


10-267952 
267649 


ll 


22 


676796 


390 


944446 


114 


73235i 


5o3 


23 


677030 


390 


944377 


114 


732653 


5o3 


267347 


n 


24 


677264 


3 89 


944309 


114 


732950 


5o3 


267045 


25 


677498 

677731 


38 9 


944241 


114 


733257 
733558 


5o3 


266743 


35 


26 


38 9 
388 


944172 


114 


5o3 


266442 


34 


27 


677964 


944104 


114 


73386o 


502 


266140 


33 


28 


678107 
678430 


388 


944o36 


114 


734162 


502 


265838 


32 


29 


388 


943967 


n4 


734463 


502 


265537 


3i 


3o 


678663 


388 


943899 


114 


734764 


002 


265236 


3o 


3i 


9-678895 


38 7 


9"94383o 


n4 


9 735o66 


502 


10-264934 


29 


32 


679128 


38 7 


943761 


114 


735367 


502 


264633 


28 


33 


679360 


38 7 


943693 


n5 


7 35668 


5oi 


264332 


27 


34 


679592 


38 7 


943624 


n5 


735969 


5oi 


264o3 1 


26 


35 


679824 


386 


943555 


u5 


736269 


00 1 


26373i 


25 


36 


68oo56 


386 


9434S6 


n5 


736570 


5oi 


26343o 


24 


37 


680288 


386 


943417 


u5 


736870 


5oi 


263i3o 


23 


38 


680519 


385 


943348 


n5 


737171 


000 


262829 


22 


39 


680750 


3S5 


943279 


u5 


737471 


5oo 


262529 


21 


40 


680982 


385 


943210 


n5 


737771 


5oo 


262229 


20 


4i 


9-68i2i3 


385 


9"943i4i 


1 15 


9.738071 


5oo 


16-261929 


JO 


42 


68i443 


384 


943072 


n5 


7383 7 i 


5oo 


261629 


10 


43 


681674 


384 


943oo3 


u5 


738671 


499 


261329 


17 


44 


681905 


384 


942934 


n5 


738971 


499 


261029 


16 


45 


682i35 


384 


942864 


ir5 


739271 


499 


260729 


i5 


46 


682365 


3 S3 


942795 


116 


739570 


499 


26o43o 


14 


8 


682595 


383 


942726 


116 


739870 


499 


26oi3o 


i3 


682825 


383 


942656 


116 


740169 


499 
498 


259S3i 


12 


49 


683o55 


383 


942587 


116 


740468 


209532 


11 


5o 


683284 


382 


942517 


116 


740767 


498 


259233 


10 


Si 


9-6835U 


382 


9-942448 


116 


9.741066 


49S 


10-258934 


8 


52 


683 7 43 


3S2 


942378 


116 


74i365 


498 


258635 


53 


683972 


332 


9423o8 


116 


741664 


498 


25S336 


7 


54 


684201 


38i 


942239 


116 


741962 


497 


258o38 


6 


55 


68443o 


33i 


942169 


116 


742261 


497 


207739 


5 


56 


684658 


38i 


942099 


116 


742559 


497 


20-441 


4 


57 


684887 


38o 


942029 • 


116 


742858 


497 


257142 


3 


58 


685n5 


38o 


941909 


116 


743io6 


497 


206844 


3 


5 9 


685343 


3 80 


94 1 889 


117 


743454 


497 


256546 


I 


60 

t 


685571 


3 80 


941819 


117 


743752 


496 


206248 





Cosine. 


1 »• 


Sine. 


D. 


Cotang. 


a 


Tang. 


1 


118 


3 














61° 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC. 47 


29 c 














150° 


/ 




Sine. 


D. 


Co?ine. 


D. 


Tang. 


D. 


Cotang. 


/ 


9-68557I 


38o 


9.941819 


117 


9.743752 


496 


io»2j6248 


60 


i 


685799 


3 79 


941749 


117 


744o5o 


496 


255930 


U 


2 


686027 


379 


941679 


117 


744348 


496 


255652 


3 


686254 


379 


941609 


117 


744645 


496 


255355 


5 7 


4 


686482 


379 

3 7 8 


94l539 


117 


744943 


496 


255o57 


56 


5 


686709 
686 9 36 


941469 
941398 


117 


745240 


496 


254760 


55 


6 


378 


117 


745538 


495 


254462 


54 


I 


68 7 i63 


3 7 8 


9 4l328 


117 


745835 


495 


254i65 


53 


687389 


378 


941258 


117 


746132 


495 


253868 


52 


9 


687616 


377 


941 187 


117 


746429 


495 


253371 


5i 


1C 


687843 


377 


941117 


117 


746726 


495 


253274 


5o 


ii 


9-688069 
688295 


377 


9-941046 


Il8 


9-747023 


494 


10-252977 


% 


12 


377 
376 


940975 


Il8 


747319 


494 


25268i 


i3 


688521 


940905 
940834 


Il8 


747616 


4 9 4 


252384 


% 


U 


688747 


376 


Il8 


7479*3 


494 


252087 


i5 


688972 


3 7 6 


940763 


Il8 


748209 
7485o5 


494 


2D1791 


45 


16 


689198 


376 


940693 


Il8 


493 


251495 


44 


\l 


689423 


375 


940622 


Il8 


748801 


493 


251199 
25090J 


43 


689648 


375 


94o55i 


Il8 


749097 


493 


42 


19 


689873 


3 7 5 


940480 


ll8 


7493o3 
749689 


493 


250607 


4i 


20 


690098 


3 7 5 


940409 


Il8 


493 


25o3u 


4o 


21 


9-690323 


374 


9«94o338 


Il8 


9- 7499 g 5 


493 


io-25ooi5 


M 


22 


690548 


374 


940267 


Il8 


750281 


492 


249719 


23 


690772 


374 


940196 


Il8 


750576 


492 


249424 


z j 


24 


690996 


374 


940125 


II 9 


750872 


492 


249128 


36 


23 


691220 


373 


94oo54 


II 9 


751167 


492 


248833 


35 


26 


691444 


3 7 3 


939982 


II 9 


751462 


492 


248538 


34 


11 


691668 


3 7 3 


93991 1 


119 


731757 


492 


248243 


33 


691892 


3 7 3 


939840 


II 9 


752o52 


491 


247948 


32 


29 


6921 1 5 


3 7 2 


939768 


119 


752347 


491 


247653 


3i 


3o 


692339 


3 7 2 


939697 


II 9 


752642 


491 


247358 


3o 


3i 


9-692562 


3 7 2 


9-939625 


II 9 


9-752937 


491 


10-247063 


29 


32 


692785 


3 7 i 


939554 


II 9 


75323i 


491 


246769 


28 


33 


693008 


3 7 i 


939482 


II 9 


753526 


491 


246474 


27 


34 


693231 


3 7 i 


939410 


119 


753820 


490 


246180 


26 


35 


693453 


3 7 i 


939339 


119 


754h5 


490 


245885 


25 


36 


693676 


370 


939267 
939193 


120 


754409 


490 


245591 


24 


3 7 


693898 


370 


120 


754703 


490 


243297 


23 


38 


694120 


370 


939123 


120 


754997 


490 


245oo3 


22 


39 


694342 


370 


939052 


120 


755291 


490 


244709 
244415 


21 


40 


694564 


36 9 


930980 


120 


755585 


489 


20 


41 


9-694786 


36 9 


9-938908 


120 


9-755878 


489 


10-244122 


\% 


42 


695007 


36 9 


9 38836 


I20 


756172 


489 


243828 


43 


695229 


36 9 
368 


938763 


120 


756465 


489 


243535 


17 


44 


695450 


938691 
938619 


120 


756759 


489 


243241 


16 


45 


695671 


368 


120 


757052 


48o 
488 


242948 


i5 


40 


695892 


368 


938547 


120 


757345 


242655 


14 


% 


6961 1 3 


368 


938475 


120 


737638 


488 


242362 


i3 


696334 


36 7 


938402 


121 


75 79 3i 


488 


242069 


12 


49 


696554 


36 7 


93833o 


121 


758224 


488 


241776 


11 


5c 


696775 


36 7 


938258 


121 


758517 


488 


24 1 483 


10 


5i 


9-696995 


367 


9-938i85 


121 


9-7588io 


488 


10-241190 


% 


52 


697210 


366 


9381 i3 


121 


759102 


487 


240898 


53 


697435 


366 


938040 


121 


759395 


487 


24o6o5 


I 


54 


697654 


366 


937967 


121 


759687 


487 


24o3i3 


55 


697874 


366 


937895 


121 


759979 


487 


240021 


5 


56 


698094 


365 


937822 


121 


760272 


487 


239728 


4 


u 


6 9 83 1 3 


365 


937749 


121 


760564 


487 


239436 


3 


6 9 8532 


365 


937676 


121 


760856 


486 


239144 


2 


59 


6987 5 1 


365 


937604 


121 


761 148 


486 


238852 


1 


6c 


698970 


364 


937531 


121 


761439 


486 


238561 



/ 


/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


11* 


>° 












60° 



48 


LOGARITHMIC SINES, 


TANGENTS, ETC. Table 


n. 


30° 














149° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


"• 


Cotang. 


60 





9-698970 


364 


9.937531 


121 


9^761439 


486 


io«23856i 


i 


699189 


364 


937458 


122 


761731 


486 


238269 


u 


2 


699407 


364 


937385 


122 


762023 


486 


237977 


3 


699626 


364 


937312 


122 


7623l4 


486 


237686 


57 


4 


699844 


363 


9 3 7 238 


122 


762606 


485 


237394 


56 


5 


700062 


363 


937165 


122 


762897 


485 


237103 


55 


6 


700280 


363 


937092 


122 


763l88 


485 


236812 


54 


I 


700498 


363 


937019 
936946 
936872 


122 


763479 


485 


236521 


53 


700716 


363 


122 


763770 


485 


23623o 


52 


9 


700933 


362 


122 


764061 


485 


235939 


5i 


10 


7on5i 


362 


936799 


122 


764352 


484 


235648 


5o 


ii 


9«7oi368 


362 


9-936725 


122 


9.764643 


484 


10-235357 


11 


12 


7oi585. 


362 


936652 


123 


764933 


484 


235o6t 
234776 


i3 


701802 


36i 


936578 


123 


765224 


484 


47 


i4 


702019 


36i 


9365o5 


123 


7655l4 


484 


234486 


46 


i5 


702236 


36i 


93643i 


123 


763803 


484 


234195 


45 


16 


702402 


36i 


936357 


123 


766090 


484 


233905 


44 


!Z 


702669 
702885 


36o 


936284 


123 


766385 


483 


2336i5 


43 


36o 


936210 


123 


766675 


483 


233325 


42 


*9 


7o3ioi 


36o 


936i36 


123 


766965 


483 


233o35 


4i 


20 


703317 


36o 


936062 


123 


767255 


483 


232745 


40 


21 


9-7o3533 


359 


9-935988 


123 


9-767545 


483 


10-232455 


3 


22 


703749 


359 


935qi4 


123 


767834 


483- 


232i66 


23 


703964 


359 


935840 


123 


768124 


482 


231876 


37 


24 


704179 
7o43g5 


359 


935766 


124 


768414 


482 


23 1 586 


36 


25 


359 


935692 


124 


768703 


482 


231297 


35 


26 


704610 


358 


9356i8 


124 


768092 


482 


23ioo8 


34 


27 


704825 


358 


935543 


124 


769281 


482 


230719 


33 


28 


703040 


358 


935469 
935393 


124 


769571 


482 


230429 


32 


29 


705254 


358 


124 


769860 


481 


23oi4o 


3i 


3o 


705469 


357 


935320 


124 


77 OI 48 


481 


229852 


3o 


3i 


g-7o5683 


35 7 


9-935246 


124 


9-770437 


481 


10-229563 


2 


32 


705898 


35 7 


935171 


124 


770726 


481 


229274 


33 


7061 1 2 


35 7 


935097 


124 


77ioi5 


481 


228985 


27 


34 


706326 


356 


935o22 


124 


77i3o3 


481 


228697 


26 


35 


706539 
706753 


356 


934048 


124 


771592 
771880 


481 


228408 


25 


36 


356 


934873 


124 


480 


228120 


24 


n 


706967 


356 


934798 


125 


772168 


480 


227832 


23 


707180 


355 


. 934723 


125 


772457 


480 


227043 


22 


39 


707393 


355 


934649 


125 


772745 


480 


227255 


21 


40 


707606 


355 


934574 


125 


773o33 


480 


226967 


20 


41 


9-707819 


355 


9-934499 


125 


9-773321 


480 


10-220679 


\l 


42 


708032 


354 


934424 


125 


773608 


479 


226392 


43 


708245 


354 


934349 


125 


773896 
774184 


479 


226104 


17 


44 


708458 


354 


934274 


125 


479 


223816 


16 


45 


708670 


354 


934199 
93412J 


125 


774471 


479 


223529 


i5 


46 


708882 


353 


125 


774759 


479 


225241 


14 


47 


709094 


353 


934048 


125 


775046 


479 


224954 


i3 


48 


709306 


353 


933973 
9338q8 


125 


775333 


479 
478 


224667 


12 


49 


709018 


353 


126 


775621 


224379 


11 


5o 


709730 


353 


933822 


126 


775908 


478 


224092 


10 


5i 


9-709941 


352 


9'933747 


126 


9.776195 
776482 


47S 


io-2238o5 





52 


7ioi53 


352 


933671 


126 


478 


2235i8 


8 


53 


710364 


352 


933596 


126 


776768 


478 


223232 


7 


54 


710575 


352 


933520 


126 


777055 


478 


222945 


6 


55 


710786 


35i 


933445 


126 


777342 


478 


222658 


5 


56 


710997 
71 1208 


35i 


93336o 
933293 


126 


777628 


477 


222372 


4 


n 


35i 


126 


777910 


477 


222085 


3 


71U19 


35i 


933217 


126 


778201 


477 


221799 

22l5l2 


2 


5 9 


711629 


35o 


933i4i 


126 


778488 


477 


1 


60 


71 1839 


35o 


933o66 


126 


778774 


477 


221226 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


1 


12( 


)° 












1 


)9° 



Table 11. LOGARITHMIC SINES, 


TANGENTS, ETC 




i9 


31° 














143° 


/ 


Sine. 


D. | 


Cosine. 


D. 


Tang. 1 


D. 


Cotang. 


60 





9.711839 


35o 


9 -933o66 


126 


9'778774 


477 


10-221226 


i 


7i2o5o 


35o 


932990 


127 


779060 


47 1 
476 


220940 


tl 


2 


712260 


35o 


932914 

9 3 2 838 


127 


779346 


220654 


3 


712469 


349 


127 


779 632 


476 


220368 


5 7 


4 


712679 


349 


932762 


127 


779918 


476 


220082 


56 


5 


712880 
713098 


349 


932685 


127 


780203 


476 


219797 


55 


6 


349 


932609 
932533 


127 


780489 


476 


219511 


54 


I 


7i33o8 


349 


127 


780775 


476 


219225 
218940 


53 


7i35i7 


348 


932457 


127 


781060 


476 


52 


9 


713726 


348 


93238o 


127 


78l346 


475 


218654 


5i 


10 


713935 


348 


9323o4 


127 


78l63l 


473 


2i836 9 


5o 


n 


9-714144 


348 


9-932228 


127 


9.781916 


475 


10-218084 


3 


12 


7U352 


347 


g32i5i 


127 


78220I 


475 


217799 


i3 


7i456i 


347 


932075 


128 


782486 


475 


217514 


47 


14 


714769 
714978 


347 


93 1 998 


128 


782771 


475 


217229 


46 


i5 


347 


931021 


128 


783o56 


475 


216944 


45 


16 


7i5i86 


347 


931845 


12* 


783341 


475 


2l665g 


44 


\l 


715394 


346 


931768 


128 


■ 783626 


474 


216374 


43 


7i56o2 


346 


931691 


128 


783910 


474 


216090 


42 


19 


715809 


346 


931614 


128 


784195 


474 


2i58o5 


41 


20 


716017 


346 


93i537 


128 


784479 


474 


2l552I 


4o 


21 


9*716224 


345 


9-931460 


128 


9-784764 


474 


io-2i5236 


is 


22 


716432 


345 


9 3i383 


128 


785048 


474 


214932 


23 


716039 


345 


93i3o6 


128 


785332 


4 7 3 


214668 


37 


24 


716846 


345 


931229 


129 


7856i6 


473 


214384 


36 


25 


717053 


345 


93i 1 52 


129 


785900 


473 


214100 


35 


26 


717259 


344 


931075 


I29 


786184 


473 


2i38i6 


34 


27 


717466 


344 


930998 


129 


786468 


473 


» 2i3532 


33 


28 


717673 


344 


930921 


I29 


786752. 


473 


213248 


32 


29 


717870 
718085 


344 


93o843 


I29 


787036 


473 


2 1 2964 


3i 


3o 


343 


930766 


I29 


787319 


472 


212681 


3o 


3i 


9-718291 


343 


9 - 9 3o688 


I29 


9.787603 


472 


10-212397 


29 


32 


718497 


343 


93061 1 


I29 


787886 


472 


212114 


20 


33 


718703 


343 


93o533 


129 


788170 


472 


2ii83o 


27 


34 


7 1 8909 


343 


93o456 


129 


788453 


472 


2ii547 


26 


35 


719'U 


342 


930378 


129 


788736 


472 


211264 


25 


36 


719320 


342 


93o3oo 


i3o 


789019 


472 


210981 


24 


u 


719525 


342 


93o223 


i3o 


789302 


471 


210698 


23 


719730 


342 


93oi45 


i3o 


789580 


471 


2 1 041 5 


22 


39 


719935 


341 


930067 


i3o 


789868 


471 


2101.32 


21 


40 


720140 


341 


929989 


i3o 


790i5i 


471 


209849 


20 


4i 


9-720345 


34i 


9-929011 

929833 


i3o 


9.790434 


471 


IO-209566 


18 


42 


720549 


34i 


i3o 


790716 


471 


209284 


43 


720754 


34o 


929755 
929677 


i3o 


790909 
791281 


47 » 


209001 


\l 


44 


720958 


34o 


i3o 


47 1 


208719 


45 


721162 


34o 


929399 


i3o 


79 1 563 


470 


208437 


i5 


46 


72 1 366 


34o 


929521 


i3o 


791846 


470 


208l 54 


14 


% 


721570 


34o 


929442 


i3o 


792128 


470 


207872 


i3 


721774 


33 9 


929364 


i3i 


792410 


470 


207590 


12 


& 


721978 


33 9 


929286 


i3i 


792692 


470 


2073o8 


11 


5o 


722181 


33 9 


929207 


i3i 


792974 


470 


207026 


10 


5i 


9-722385 


33 9 


9-929129 


i3i 


9-793256 


470 


IO-206744 


I 


52 


722588 


33 9 
338 


929050 


i3i 


793538 


469 


206462 


53 


722791 


928972 


i3i 


793819 


469 


206l8l 


1 


54 


722994 


338 


928893 


i3i 


794101 


469 


205899 


6 


55 


723i 97 


338 


928815 


i3i 


794383 


469 


2o56i7 


5 


56 


723400 


338 


928736 


l3i 


794664 


469 


2o5336 


4 


u 


7236o3 


337 


928657 
928378 


i3i 


794946 


469 


. 2o5o54 


3 


7238o5 


33 7 


i3i 


790227 


4&9 


204773 


2 


5 9 


724007 


337 


928499 


i3i 


7955o8 


468 


204492 


1 


60 


1 724210 


33 7 


928420 


i3i 


795789 


468 


2042 1 1 




! 


' 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


121 















58° 



40 



60 


LOGARITHMIC SINES, 


TANGENTS, ETC. Table II. 


32° 

r 














147° 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


t 

60 





9«7242I0 


33 7 


9.928420 


1 32 


9.795789 


468 


IO'2042II 


I 


724412 


33 7 


928342 


132 


79607c 


468 


203930 


% 


2 


724614 


336 


Q28263 


132 


79635l 


468 


2o3649 

2o3368 


3 


724816 


336 


928l83 


132 


796632 


468 


57 


4 


725oi7 


336 


928104 


132 


796913 


468 


203087 
202806 


56 


5 


725219 


336 


928025 


132 


797194 


468 


55 


6 


725420 


335 


927946 


i3? 


797474 


468 


202526 


54 


I 


725622 


335 


927867 


132 


797755 


468 


202245 


53 


725823 


335 


927787 


132 


798o36 


467 


201964 


52 


9 


726024 


335 


927708 


132 


7983l6 


467 


201684 


5i 


10 


726225 


335 


927629 


132 


798596 


467 


201404 


5o 


ii 


9-726426 


334 


9-927549 


132 


9.798877 
7 99 l5 7 


467 


10-201123 


% 


12 


726626 


334 


927470 


1 33 


467 


200843 


i3 


726827 


334 


92739O 


i33 


799437 


467 


2oo563 


% 


i4 


727027 


334 


927310 


i33 


799717 


467 


200283 


i5 


727228 


334 


927231 


i33 


799997 


466 


200003 


45 


16 


727428 


333 


927l5l 


i33 


800277 


466 


199723 


44 


n 


727628 


333 


92707I 


i33 


8oo557 


466 


199443 


43 


18 


727828 


333 


92699I 


i33 


8oo836 


466 


199164 


42 


*9 


728027 


333 


92691 1 
926831 


1 33 


801 1 16 


466 


198884 


4i 


20 


728227 


333 


i33 


801396 


466 


198604 


40 


21 


9-728427 

728626 


332 


9-926751 


i33 


9-801675 


466 


10-198325 


ll 


22 


332 


926671 


i33 


801955 


466 


198045 


23 


728825 


332 


92659I 


1 33 


802234 


465 


197766 


37 


24 


729024 


332 


9265 1 1 


1 34 


8o25i3 


465 


197487 
197208 


36 


25 


729223 


33i 


92643l 


1 34 


802792 


465 


35 


26 


729422 


33i 


92635i 


i34 


803072 


465 


196928 


34 


3 


729621 


33i 


926270 


i34 


8o335i 


465 


196649 


33 


729820 


33i 


926190 


1 34 


8o363o 


465 


196370 


32 


29 


730018 


33o 


9261 10 


1 34 


803909 


465 


1 9609 1 


3i 


3o 


730217 


33o 


926029 


1 34 


804187 


465 


1 938 1 3 


3o 


3i 


9-73o4i5 


33o 


9-925949 

925868 


134 


9-804466 


464 


10-195534 


3 


32 


73o6i3 


33o 


i34 


8o4745 


464 


190255 


33 


73o8i 1 


33o 


925788 


i34 


8o5o23 


464 


194977 
194698 


27 


34 


731009 


329 


923707 


i34 


8o53o2 


464 


26 


35 


731206 


329 


925626 


1 34 


8o558o 


464 


194420 


25 


36 


731404 


329 


925545 


i35 


8o5859 


464 


19-I141 


24 


3t 


731602 


329 


925465 


i35 


806137 


464 


193863 


23 


38 


731799 


32g 

328 


925384 


i35 


80641 5 


463 


1 9 3 585 


22 


3 9 


731996 


9253o3 


i35 


806693 


463 


193307 


2! 


4o 


732193 


328 


925222 


i35 


806971 


463 


193029 


20 


4i 


9-732390 


328 


9-925141 


i35 


9-807249 


463 


10-192751 


% 


42 


732587 


328 


925060 


i35 


807527 


463 


192473 


43 


732784 


328 


924079 


i35 


807805 


463 


192195 


17 


44 


732980 


32 7 


924897 


i35 


8o8o83 


463 


191917 


16 


45 


733177 


32 7 


924816 


i35 


8o836i 


463 


191639 


i5 


46 


7333 7 3 


32 7 


924735 


1 36 


8o8638 


462 


191362 


14 


% 


733569 
733 7 65 


32 7 


924654 


1 36 


808916 


462 


191084 


i3 


3 27 


9*24572 


i36 


809 1 93 


462 


190807 


12 


4 9 


733961 


326 


924491 


i36 


80947 1 


462 


190529 


11 


5o 


734i57 


326 


924409 


i36 


809748 


462 


190202 


10 


5i 


9.734353 


326 


9-924328 


1 36 


9-810025 


462 


10-189975 


I 


52 


734549 


326 


924246 


i36 


8io3o2 


462 


189698 


53 


734744 


325 


924164 


i36 


8io58o 


462 


1S9420 


7 


54 


73493Q 


325 


924083 


1 36 


810857 


462 


1 891 43 
188866 


6 


55 


73di35 


325 


924001 


i36 


8iu34 


461 


5 


56 


73533o 


3a5 


923919 


1 36 


811410 


461 


iSS5 9 o 


4 


57 


735525 


325 


923837 


1 36 


81 1687 


461 


i883i3 


3 


58 


735719 


324 


923755 


l3 7 


81 1964 


461 


i88o36 


2 


5 9 


735914 


324 


923673 


:3 7 


812241 


461 


187750 


1 


60 


736109 


324 


923591 


i3 7 


812D17 


461 


187483 




1 


1 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


125 


>0 












P7° 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC 


61 


88° 














146° 


t 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 1 ' 





9-736ioo 
7363o3 


324 


9-923591 


i3 7 


9-8i25i7 


46l 


10-187483 


60 


i 


324 


923509 


i3 7 


812794 


461 


187206 


n 


2 


736498 


324 


923427 
923345 


i3 7 


813070 


46l 


186930 


3 


736692 
736886 


323 


i3 7 


8i3347 


46o 


186653 


57 


4 


323 


923263 


i3 7 


8i3623 


460 


186377 


56 


5 


737080 


323 


923i8i 


i3 7 


813899 


460 


186101 


55 


6 


737274 


323 


923098 


i3 7 


814176 
8i4452 


46o 


185824 


54 


i 


737467 


323 


923oi6 


i3 7 


460 


185548 


53 


8 


737661 


322 


922o33 
922851 


i3 7 


814728 


460 


185272 


52 


9 


737855 


322 


i3 7 
i38 


8i5oo4 


460 


184996 


5i 


10 


738048 


322 


922768 


8i528o 


460 


184720 


5o 


ii 


9*738241 


322 


9-922686 


1 38 


g-8i5555 


459 


io.i84445 


% 


12 


738434 


322 


922603 


i38 


8i583i 


459 


184169 
i838 9 3 


i3 


738627 


321 


922520 


i38 


816107 


459 


47 


14 


738820 


321 


922438 


i38 


8i6382 


459 


i836i8 


46 


i5 


739013 


321 


922355 


i38 


8 16658 


459 


183342 


45 


16 


739206 


321 


922272 
922189 


i38 


8i6 9 33 


459 


183067 


44 


17 


739398 


321 


138 


817209 


459 


182791 
182516 


43 


18 


739390 


320 


922106 


138 


817484 


459 


42 


*9 


739783 


320 


922023 


i38 


817759 
8i8o35 


459 
458 


182241 


4i 


20 


739975 


320 


921940 


i38 


i8i 9 65 


4o 


21 


9-740167 


320 


9-921857 


1 39 


9-8i83io 


458 


10-181690 


1% 


22 


740359 


320 


921774 


139 


8i8585 


458 


i8i4i5 


23 


74o55o 


319 


921691 


1 39 


818860 


458 


181140 


u 


24 


740742 


319 


921607 


139 


819135 


458 


i8o865 


25 


740934 


319 


921524 


i3g 


819410 


458 


180590 


35 


26 


74H25 


319 


921441 


1 39 


819684 


458 


i8o3i6 


34 


11 


74i3i6 


3m 


921357 


1 39 


819959 


458 


180041 


33 


741 5o8 


3i8 


921274 


l3 9 


820234 


458 


179766 


32 


29 


741699 

741889 


3i8 


92 1 190 


139 


82o5o8 


45 7 


179492 


3i 


3o 


3i8 


921107 


139 


820783 


457 


179217 


3o 


3i 


9-742080 


3i8 


9-921023 


1 39 


9-821057 


457 


10-178943 


i 


32 


742271 


3i8 


920939 


140 


82i332 


457 


178668 


33 


742462 


3.7 


920806 


140 


821606 


457 


178394 


27 


34 


7426D2 


3i7 


920772 


140 


82188c 


457 


178120 


26 


35 


742842 


3i 7 


920688 


140 


822154 


457 


177846 


25 


36 


743o33 


3i 7 


920604 


1 4o 


822429 
82270J 


457 


177571 


24 


n 


743223 


3i 7 


920520 


140 


457 


177297 


23 


7434i 3 


3i6 


920436 


140 


822977 


456 


177023 


22 


39 


7436o2 


3i6 


92o352 


140 


823251 


456 


176749 


21 


4o 


743792 


3i6 


920268 


140 


823524 


456 


176476 


20 


4i 


9-743982 


3i6 


9-920184 


140 


9.823798 


456 


10-176202 


\i 


42 


744171 


3i6 


920099 
920016 


140 


824072 


456 


175928 


43 


744361 


3i5 


140 


824345 


456 


175655 


\l 


44 


74455o 


3i5 


9 1 903 1 
919846 


141 


824619 
824893 


456 


I7538i 


45 


74473o 
744928 


3i5 


141 


456 


175107 


ID 


46 


3i5 


919762 


141 


825i66 


456 


174834 


14 


47 


745ii7 


3i5 


919677 


141 


825439 
825713 


455 


I7456i 


i3 


48 


7453o6 


3i4 


919593 


Ui 


455 


174287 


12 


49 


7454Q4 
' 745683 


3i4 


919508 


141 


825986 


455 


1 740 1 4 


11 


5o 


3i4 


919424 


141 


826259 


455 


I7374I 


10 


5i 


9.745871 


3i4 


9-919339 


Ui 


9-826532 


455 


10-173468 


8 


52 


746060 


3i4 


919254 


141 


826805 


455 


173195 


53 


746248 


3i3 


919169 


141 


827078 


455 


172922 
172649 


7 


54 


746436 


3i3 


919083 


Ui 


827351 


455 


6 


55 


746624 


3i3 


919000 


Hi 


827624 


455 


172376 


5 


56 


746812 


3i3 


9 1 89 1 5 

9 i883o 


142 


827897 
828170 


454 


172103 


4 


57 


746999 


3i3 


142 


454 


171830 


3 


58 


747187 


3l2 


918745 


142 


828442 


454 


i 7 i558 


2 


5 9 


747374 


3l2 


918659 


142 


858715 


454 


171283 


1 


6o 
/ 
12i 


747562 


3l2 


918574 


142 


828987 


454 


171013 





Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. | / 


1° 












66° 



52 


LOGARITHMIC SINES, 


TANGENTS, ETC. Table 


IL 


34 c 














145° 


/ 


Sine. 


D. 


Cos. ne. 


D. 


Tang. 


D. 


Cotang. 


/ 





9-747062 


312 


Q^9l8574 


142 


9-828987 


454 


I0-I7I0I3 


60 


I 


747749 


3l2 


918489 


142 


829260 


454 


170740 


It 


2 


7479^6 


312 


918404 


142 


82o532 


454 


I70468 


3 


748123 


3u 


oi83i8 


142 


829803 


454 


170195 


57 


4 


7483io 


3u 


9i8233 


142 


830077 


454 


169923 


56 


5 


748497 


3n 


918147 


142 


83o349 


453 


169651 


55 


6 


748683 


3n 


918062 


142 


83o62i 


453 


169379 


54 


I 


748870 


3u 


917976 
917891 


143 


83o8 9 3 


453 


169107 


53 


749056 


3io 


143 


83n65 


«453 


168835 


52 


9 


749243 


3io 


917805 


143 


83i437 


453 


168563 


5i 


10 


749429 


3io 


917719 


143 


831709 


453 


168291 


5o 


II 


9-749615 


3io 


9.917634 


143 


9-831981 


453 


10-168019 


8 


12 


749801 


3io 


917348 


143 


832253 


453 


167747 


i3 


7499S7 


3 09 


917462 


143 


832525 


453 


167475 


% 


14 


750172 


309 


917376 


143 


832796 


453 


167204 


i5 


75o358 


309 


917290 


143 


833o68 


452 


166932 


45 


16 


73o543 


309 


917204 


143 


83333 9 


452 


1 6666 1 


44 


17 


750729 


309 
3o8 


917118 


144 


8336 1 1 


452 


i6638o 


43 


18 


750914 


917032 


144 


833882 


452 


166118 


42 


19 


751099 
751284 


3o8 


916046 


144 


834154 


452 


165846 


4i 


20 


3o8 


916859 


144 


834425 


452 


165575 


4o 


21 


9-751469 


3o8 


9-916773 


144 


9-834696 


452 


io-i653o4 


S 


22 


75i654 


3o8 


916687 


144 


834967 


452 


i65o33 


23 


731839 


3o8 


916600 


144 


835238 


452 


164762 


37 


24 


752023 


307 


9i65i4 


144 


835509 


452 


1 6449 1 


36 


25 


752208 


307 


916427 


144 


835780 


45i 


164220 


35 


26 


752392 


307 


916341 


144 


836o5i 


45 1 


163949 
163678 


34 


27 


732576 


307 


916254 


144 


836322 


45i 


33 


28 


752760 


307 


916167 


U5 


836393 


45 1 


163407 


32 


29 


732944 


3o6 


916081 


145 


836864 


45i 


i63i36 


3i 


3o 


753128 


3o6 


913994 


145 


837i34 


45 1 


162866 


3o 


3i 


9-7533i2 


3o6 


9- 91 5oo 7 
913820 


145 


9-8374o5 


45i 


10-162595 


3 


32 


753495 


3o6 


145 


837675 


45 1 


162325 


33 


733679 


3o6 


915733 


145 


837946 


45i 


162054 


27 


34 


753862 


3o5 


913646 


145 


8382i6 


45i 


161784 


26 


35 


754046 


3o5 


913559 


145 


838487 


45o 


i6i5i3 


25 


36 


734229 


3o5 


915472 


145 


838757 


45o 


161243 


24 


37 


754412 


3o5 


91 5385 


145 


839027 


45o 


160973 


23 


38 


704593 


3o5 


915297 


145 


839297 


45o 


160703 


22 


3 9 


754778 


3o4 


915210 


145 


83 9 568 


45o 


160432 


21 


4o 


754960 


3 04 


9i5i23 


146 


83 9 838 


45o 


160162 


20 


4i 


9-755i43 


3o4 


9"9i5o35 


146 


9-840108 


45o 


10-159892 


3 


42 


755326 


3 04 


9U948 


146 


840378 


45o 


159622 


43 


7555o8 


3o4 


914860 


146 


840648 


45o 


159352 


17 


44 


755690 


3 04 


914773 


146 


840917 


449 


139083 


16 


45 


755872 


3o3 


914683 


146 


841187 


449 


i5S8i3 


i5 


46 


756o54 


3o3 


914598 


146 


841457 


449 


1 58543 


14 


a 


756236 


3o3 


9i45io 


146 


841727 


449 


i5S2 7 3 


i3 


756418 


3o3 


914422 


146 


841996 


449 


i5Soo4 


12 


49 


756600 


3o3 


914334 


146 


842266 


449 


i5 7 734 


1 1 


5o 


756782 


302 


914246 


147 


842535 


449 


1 57465 


10 


5i 


9-756963 


302 


9«9i4i58 


147 


9-842805 


449 


10-157195 


I 


52 


757144 


302 


914070 


147 


843074 


449 


156926 


53 


757326 


302 


913982 
9 i3S 9 4 


147 


843343 


449 


106057 


7 


54 


757507 


302 


147 


8436i2 


449 


156388 


6 


55 


757688 


3oi 


9i38o6 


147 


843882 


448 


1 56 1 1 8 


5 


56 


757869 


3oi 


913718 


147 


844i5i 


448 




4 


57 


758o5o 


3oi 


9i363o 


147 


844420 


448 


i5558o 


3 


58 


75823o 


3oi 


9i354i 


147 


844689 
844958 


448 


1 553 1 1 


: 


5 9 


75841 1 


3oi 


9i3453 


147 


448 


i55o42 


1 


60 


758591 


3oi 


9i3365 


147 


845227 


44S 


154773 





/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


* 


Tang. 


55° j 


124 


















Table II. LOGARITHMIC SINES, 


TANGENTS, ETC 


!. 68 


86° 














114° 


/ 


Sine. 


u 1 


Cosine. 


D. j 


Tang. 


D. 


Cotang. 


/ 





g-7585gi 


3oi 


9-9i3365 


147 


9-845227 
845496 


448 


10-154773 


60 


i 


7 58 77 2 


3oo 


913276 


148 


448 


I 545o4 


u 


2 


758932 


3oc 


913187 


845764 


448 


154236 


3 


759l32 


3oo 


913099 


148 


846o33 


448 


153967 


57 


4 


759312 


3oo 


9i3oio 


148 


846302 


448 


1 536o8 


56 


5 


759492 


3oo 


912922 


148 


846570 


447 


1 534oo 


55 


6 


759672 


299 


912833 


148 


846839 


447 


i53i6i 


54 


I 


759852 


299 


912744 


148 


847108 


447 


152892 


53 


76oo3l 


299 


912655 


148 


847376 


447 


152624 


5a 


9 


7602 1 1 


299 


912566 


148 


847644 


447 


i5 2 356 


5i 


10 


760390 


299 


912477 


148 


847913 


447 


152087 


5o 


ii 


9« 760569 
760748 


298 


9912388 


148 


9.848181 


447 


io-i5i8i9 


% 


12 


298 


912299 


149 


848449 


447 


i5i55i 


i3 


760927 
761 106 


298 


912210 


149 


8 J s c,m 


447 


i5i283 


47 


14 


298 


912121 


149 


848986 


447 


i5ioi4 


46 


i5 


761285 


298 


9i2o3i 


149 


849254 


447 


150746 


45 


16 


761464 


298 


91 1942 
9 ii853 


149 


849522 


447 


150478 


44 


«7 


761642 


297 


149 


849790 

85ooD7 
85o325 


446 


I502IO 


43 


18 


761821 


297 


91 1763 


149 


446 


149943 


42 


19 


761999 


297 


91 1674 


149 


446 


149675 


4i 


20 


762177 


297 


91 1 584 


149 


85o593 


446 


149407 


4o 


21 


9-762356 


297 


9-911495 


149 


9-85o86i 


446 


10-149139 

I 48871 


% 


22 


762534 


296 


9ii4o5 


149 


85 1 129 


446 


23 


762712 


296 


91 i3i5 


i5o 


85 1 396 


446 


148604 


37 


24 


762889 


296 


911226 


i5o 


85 1664 


446 


148336 


36 


25 


763067 
763245 


296 


9iii36 


i5o 


85i 9 3i 


446 


148069 


35 


26 


296 


91 1046 


i5o 


852199 


446 


147801 


34 


27 


763422 


296 


910956 


i5o 


852466 


446 


147534 


33 


28 


763600 


295 


910866 


i5o 


852 7 33 


445 


147267 


32 


29 


763777 


295 


910776 


i5o 


853ooi 


445 


146999 
146732 


3i 


3o 


763954 


295 


910686 


i5o 


853268 


445 


3o 


3i 


9-764131 


295 


9-9io5g6 


i5o 


9-853535 


445 


IO-I46465 


3 


32 


764308 


295 


9io5o6 


i5o 


853802 


445 


146198 
145901 


33 


764485 


294 


9io4i5 


i5o 


854069 


445 


27 


34 


764662 


294 


9io325 


i5i 


854336 


445 


145664 


26 


35 


764838 


294 


910235 


i5i 


8546o3 


445 


145397 
i45i3o 


25 


'36 


765oi5 


294 


910144 


i5i 


854870 


445 


24 


37 


765191 


294 


910064 


i5i 


855i37 


445 


U4863 


23 


33 


765367 


294 


909963 


i5i 


8554o4 


445 


144596 


22 


39 


765544 


293 


909873 
909782 


i5i 


8556 7 i 


444 


144329 


21 


4o 


760720 


293 


i5i 


855 9 38 


444 


144062 


20 


4i 


9-765896 


293 


9-909691 


i5i 


9-856204 


444 


10-143796 


\% 


42 


766072 


293 


909601 


i5i 


856471 


444 


143529 


43 


766247 


293 


909510 


i5i 


856737 


444 


143263 


17 


44 


766423 


293 


909419 


i5i 


857004 


444 


142996 
142730 


16 


45 


766598 


292 


909328 


l52 


857270 


444 


i5 


46 


766774 


292 


909237 


I 52 


85 7 5:»7 
8578&J 


444 


142463 


14 


47 


766949 


292 


909146 


I 52 


444 


142197 
141931 


i3 


48 


767124 


292 


909055 


1 52 


858069 


444 


12 


49 


767300 


292 


908964 


I 52 


858336 


444 


141664 


11 


5o 


767475 


291 


908873 


l52 


858602 


443 


141398 


10 


5i 


9.767649 


291 


9-908781 


l52 


y- 858868 


443 


io-i4ii32 


I 


52 


767824 


291 


908690 


l52 


85gi34 


443 


140866 


53 


767990 
768173 


291 


908599 


l52 


859400 


443 


140600 


7 


54 


291 


908507 
908416 


1 52 


859666 


443 


140334 


6 


55 


768348 


290 


1 53 


859932 


443 


140068 


5 


56 


768522 


290 


908324 


1 53 


860198 


443 


139802 


4 


a 


768697 


290 


908233 


1 53 


860464 


443 


i39536 


3 


768871 


290 


. 908141 


1 53 


860730 


443 


139270 


2 


59 


769045 


290 


908049 
907958 


1 53 


860995 


443 


139005 


1 


60 


769219 


290 


1 53 


861261 


443 


138739 





/ 


Cosine. 


1 * 


, Sine. 


D. 


Cotang. 


D. 


Tang. 1 / 


12, 


)° 












64° 



54 


LOGARITHMIC SINES, 


TANGENTS, ETC 


Table LT. 1 


36° 














143° 


/ 


Sine. 


D. | 


Cosine. 


D. | 


Tang. 1 


D. 


Cotang. / 





9.769219 
769393 


29O 

289 


9'9C7o58 


153 


9-861261 


443 


10-138739 
138473 


60 


i 


907866 


1 53 


86l 527 


443 


S3 


2 


769566 


289 


907774 


1 53 


861792 


442 


1 38208 


3 


769740 


289 


907682 


1 53 


862058 


442 


1 3 79 42 


57 


4 


769913 


289 


907590 


i53 


862323 


442 


i3 7 6 77 


56 


5 


770087 


289 
288 


907498 


1 53 


862589 


442 


137411 


55 


6 


770260 


907406 


1 53 


862854 


442 


137146 


54 


I 


770433 


288 


907314 


1 54 


363ii9 


442 


i3688i 


53 


770606 


288 


907222 


1 54 


863385 


442 


1 366i 5 


52 


9 


770779 


288 


007129 


104 


86365o 


442 


i3635o 


5i 


10 


770902 


288 


907037 


144 


863 9 i5 


442 


i36o85 


30 


ii 


9 77II25 


288 


9-906045 
906852 


154 


3.864180 


442 


io-i3582o 


S 


12 


771298 


287 


1 54 


864445 


442 


i35555 


i3 


77U70 


287 


906760 


104 


864710 


442 


i352oo 


47 


U 


771643 


287 


906667 


1 54 


864975 


441 


i35o25 


46 


i5 


771810 


287 


906570 


1 54 


865240 


44i 


134760 


45 


16 


771987 


287 


906482 


154 


8655o5 


44i 


134495 


44 


H 


772169 


287 


906389 


1 55 


865770 


441 


1 3423o 


43 


18 


77233l 


286 


906296 


1 55 


866o35 


441 


i33o65 


42 


l 9 


7725o3 


286 


906204 


i55 


8663oo 


441 


133700 


4i 


20 


772675 


286 


906111 


i55 


866564 


441 


133436 


4o 


21 


9.772847 
773018 


286 


9-906018 


i55 


9.866829 


44i 


io-i33i7i 


* 


22 


286 


905925 


1 55 


867094 


441 


132906 


23 


773190 


286 


905832 


i55 


867358 


441 


132642 


37 


24 


77336i 


285 


905739 


1 55 


867623 


441 


132377 


36 


25 


773533 


285 


905645 


1 55 


867887 


44i 


i32ii3 


35 


26 


773704 


285 


905552 


i55 


8681 52 


440 


i3i848 


34 


27 


7738 7 5 


285 


905459 


1 55 


868416 


44o 


i3i584 


33 


28 


774046 


285 


9o5366 


1 56 


868680 


440 


i3i32o 


32 


29 


774217 
774388 


285 


905272 


i'56 


868940 


44o 


i3io5o 


3i 


3o 


284 


905179 


106 


869209 


44o 


130791 


3o 


3i 


9-774558 


284 


9-oo5o85 


1 56 


9-869473 


44o 


io>i3o527 


3 


32 


774729 


284 


904992 


1 56 


869737 


44o 


i3o263 


33 


774899 


284 


904898 


1 56 


870001 


44o 


129999 


27 


34 


770070 


284 


904804 


i56 


870265 


44o 


129733 


26 


35 


775240 


284 


9047 1 1 


1 56 


870529 
870793 


440 


129471 


25 


36 


775410 


283 


904617 


1 56 


44o 


1 29207 


24' 


37 


77558o 


283 


904523 


1 56 


871057 


44o 


128943 


23 


38 


775750 


283 


904429 


i5 7 


871321 


44o 


128679 


22 


39 


770920 


283 


904335 


i5 7 


87i585 


440 


128413 


21 


4o 


776090 


283 


904241 


i5 7 


871849 


43 9 


128101 


20 


4i 


9.776259 


283 


9-904147 


i5 7 


9-872112 


43 9 


10-127888 


\t 


42 


776429 
776098 


282 


9o4o53 


107 


872376 


43 9 


127624 


43 


282 


903909 
903864 


107 


872640 


439 


127360 


\l 


44 


776768 


282 


167 


872903 


439 


127007 
126833 


45 


776937 


282 


903770 


i5 7 


873167 


43 9 


i5 


46 


777106 


282 


903676 


i5 7 


87343o 


439 


126570 


14 


47 


777275 


281 


903 58 1 


167 


873694 


43 9 


i263o6 


i3 


48 


777444 


281 


903487 


i5 7 


873907 


43 9 


1 26o43 


12 


49 


777613 


281 


903392 


158 


874220 


43 9 


120780 


11 


5o 


777731 


581 


903298 


i58 


874484 


43 9 


i25oi6 


10 


5i 


9-777950 


281 


9'9o32o3 


1 58 


9.874747 


43 9 


10-120253 


1 


52 


778119 


281 


903 108 


i58 


875010 


439 


12^990 


53 


778287 


280 


9o3oi4 


1 58 


875273 


438 


124727 


7 


54 


778405 


280 


902919 
902024 


i58 


875537 


433 


124463 


6 


55 


778624 


280 


1 58 


875800 


438 


124200 


5 


56 


778792 


280 


902729 


1 58 


876063 


438 


i23o37 


4 


57 


778960 


280 


902634 


1 58 


876326 


433 


123674 


3 


58 


779128 


280 


902539 


1 59 


876589 


438 


123/, II 


2 


59 


779290 


279 


902444 


109 


876852 


438 


I23i48 


1 


6o 

/ 


779463 


279 


902349 


159 


8771 14 


438 


122S86 





Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


1 


I2< 


i° 












68° 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC. 65 


37° 














142° 





Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


t 


9-779463 


279 


9.902349 


1 59 


9'877U4 


438 


10-122886 


60 


i 


779631 


279 


90225J 


159 


877377 


438 


122623 


ll 


2 


770798 


279 


902 1 58 


1 59 


877640 


438 


122360 


3 


779966 


279 


902063 


159 


877903 
878165 


438 


122097 
I2i835 


5 7 


4 


780133 


279 


901067 
901872 


l59 


438 


56 


5 


780300 


278 


159 


878428 


438 


1 21672 


55 


6 


780467 . 


278 


901776 


l59 


87869I 
878963 


438 


121309 


54 


I 


780634 


278 


901681 


1 59 


437 


121047 


53 


780801 


278 


90 i 585 


1 59 


879216 


437 


120784 


52 


9 


780968 


^ 


901490 


109 


879478 


437 


120522 


5i 


10 


781134 


278 


901394 


160 


879741 


43 7 


120259 


5o 


ii 


9-781301 


277 


9.901298 


160 


9<88ooo3 


437 


IO.II9997 


% 


12 


781468 


277 


901202 


160 


880205 


437 


1 19735 


1 3 


781634 


277 


901 106 


160 


88o528 


437 


119472 


47 


U 


781800 


277 


901010 


160 


880790 


437 


II92IO 


46 


i5 


781966 


277 


900914 
9008 1 8 


160 


88io52 


437 


I18948 


45 


16 


782132 


277 


160 


88i3i4 


437 


1 1 8686 


44 


\l 


782298 


276 


900722 


160 


881577 


437 


1 1 8423 


43 


782464 


276 


900626 


160 


881 83 9 


437 


118161 


42 


*9 


782630 


276 


900529 
900433 


160 


882101 


437 


1 17899 
117637 


4i 


20 


782796 


276 


161 


882363 


436 


4o 


21 


9.782961 


276 


9.900337 


161 


9.882625 


436 


10.117375 


It 


22 


783127 


276 


900240 


161 


882887 
883 1 48 


436 


1 171 13 


23 


783292 
783458 


275 


900144 


161 


436 


1 i6852 


37 


24 


2 7 5 


900047 
8999? 1 
8 99 854 


161 


8834io 


436 


1 1 6590 


36 


25 


783623 


275 


161 


8836 7 2 


436 


1 16328 


35 


26 


783788 


275 


161 


883 9 34 


436 


1 1 6066 


34 


S 


783953 


275 


899757 


161 


884196 


436 


n58o4 


33 


7841 18 


275 


899660 


161 


884457 


436 


1 1 5543 


32 


29 


784282 


274 


899564 


161 


884719 


436 


ii528i 


3i 


3o 


784447 


274 


899467 


162 


884980 


436 


115020 


3o 


3i 


9.784612 


274 


9-899370 


162 


9-885242 ' 


436 


io- 1 14758 


3 


32 


784776 


274 


899273 


162 


8855o4 


436 


1 1 4496 


33 


784941 


274 


899176 


162 


88D765 


436 


1 14235 


27 


34 


785io5 


274 


899078 


162 


886026 


436 


1 13974 


26 


35 


785269 


2 7 3 


898981 
898884 


162 


886288 


436 


113712 


25 


36 


785433 


2 7 3 


162 


886549 


435 


11 345 1 


24 


37 


785597 


2 7 3 


898787 


162 


8868 1 1 


4 35 


Ii3i8g 


23 


33 


785761 


2 7 3 


898689 


162 


887072 


435 


1 1 2928 


22 


3 9 


785 9 25 


273 


8 9 85 9 2 


162 


887333 


435 


1 1 2667 


21 


4o 


786089 


2 7 3 


898494 


1 63 


887594 


435 


1 1 2406 


20 


4i 


9-786252 


272 


9-898397 


1 63 


9-887855 
8881 16 


435 


io«ii2i45 


I? 


42 


786416 


272 


898299 


1 63 


435 


1 1 1884 


43 


786579 


272 


898202 


1 63 


8883 7 8 


435 


111622 


\l 


44 


786742 


272 


898104 


1 63 


888639 


435 


iii36i 


45 


786906 


272 


898006 


1 63 


888900 


435 


1 moo 


i5 


46 


787069 


272 


897908 
897810 


1 63 


889161 


435 


1 10839 


14 


An 


787232 


271 


1 63 


889421 


435 


1 10579 
no3i8 


i3 


48 


7 8 7 3o5 


271 


897712 


1 63 


889682 


435 


12 


49 


787537 


271 


897614 


1 63 


889943 


435 


1 10057 


11 


5o 


787720 


271 


897516 


1 63 


890204 


434 


109796 


10 


5i 


9.787883 


271 


9-897418 


164 


9 • 890465 


434 


10-109535 


8 


52 


788045 


271 


897320 


164 


890725 


434 


109275 


53 


788208 


271 


897222 


164 


890986 


434 


1 0901 4 

108753 


7 


54 


788370 


27O 


897123 


164 


891247 


434 


6 


55 


788532 


270 


897025 


164 


891507 


434 


108493 


5 


56 


788694 
788856 


27O 


896926 
896828 


164 


891768 


434 


108232 


4 


n 


270 


164 


892028 


434 


107972 


3 


789018 


270 


806729 


164 


892289 


434 


107711 


2 


5 9 


789180 


270 


896631 


164 


892549 


434 


107451 


1 


60 


789342 


269 


896532 


164 


892810 


434 


107190 




r 


t 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


12' 


r° 














52° 



56 


LOGARITHMIC SINES, 


TAXGEXT 


S, ETC 


Table H. 


38° 














141° 


/ 


S^e. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. | ' 





9789342 


269 


9-896532 


164 


9-Sq2Sic 


434 


IO-IO-igo 60 


i 


-:cO04 


269 ; 


896433 


i65 


8930-0 


434 


I06930 


Si 


2 


789665 


269 


896335 


IOD 


893331 


434 


1:0:00 


3 


789827 


269 


S 00236 


i65 


8 9 35 9 l 


434 


106409 


57 


4 


759988 


269 


8061 3- 
896038 


i65 


893801 


434 


I06U9 


56 


5 


790149 


269 
268 


l65 


8941 1 1 


434 


:: '::: 


55 


6 


7903 10 


&9D939 


165 


8943-2 


434 


io5628 


04 


8 


7904-1 


2:5 


895^40 


i65 


8 9 463 2 


433 


io5368 


53 


790632 


268 


89D-4I 


i65 


894592 


433 


io5jo8 


52 


9 


790793 


268 


S95641 


i65 


&95 1 52 


433 


104543 


01 


10 


7909D4 


268 


895542 


i65 


895412 


433 


: :_: :o 


00 


ii 


9 ■ -91 1 1 5 


268 


9'?9 5 443 


166 


9-8956-2 


433 


10-104328 


% 


12 


7912-D 


267 


895343 


166 


895932 


433 


10406S 


i3 


791436 


20- 


895244 


166 


896192 


433 


::] :5 


% 


14 


-91D96 


26 7 


895145 


166 


896452 


433 


io3548 


ID 


79 I_5 7 


267 


895045 


166 


896-712 


433 


::::: 


45 


16 


79*9*7 


267 


894945 


166 


8969-1 


433 


103029 


44 


17 


-020-7 


267 


894*46 


166 


89-231 


433 


102-09 


43 


18 


-12237 


266 


894-46 


166 


89-491 


433 


102009 


42 


] 9 


792:;- 


266 


894646 


166 


897751 


433 


. 02249 


41 


20 


792DD7 


266 


894546 


166 


89SOIO 


433 


101990 


4o 


21 


9-792716 


266 


9 • ?o4446 


167 


9-898270 


433 


10-101730 


\% 


22 


' 792876 


266 


894346 


167 


: -53o 


433 


1014-0 


23 


793o35 


266 


894246 


167 


0:0-59 


433 


101211 


37 


24 


793195 


265 


894U6 


167 


899049 
899308 


432 


100901 


36 


23 


793354 


265 


894046 


167 


432 


10:002 


35 


26 


79 35l 4 


2o5 


893946 


167 


::o563 


432 


1 0043 2 


34 


2" 


793673 


205 


893^46 


167 


890:1- 


432 


100173 


33 


28 


79383a 


265 


893-45 


16- 


900087 


432 


099913 


32 


29 


793991 


265 


893o_: 


167 


900346 


432 


099604 


3i 


3o 


794i 5o 


264 


893544 


167 


900600 


432 


099390 


3o 


3i 


9- _ 943o8 


264 


9-8o3444 


168 


9 • 900864 


432 


10-099136 


3 


32 


-:446 7 


264 


893343 


16S 


901124 


432 


: - - - : 


33 


794626 


264 


893243 


168 


90i353 


432 




2" 


34 


"94-^4 


264 


893142 


168 


901642 


432 


09835a 


20 


3d 


794942 


204 


8 9 3o4i 


100 


901901 


432 


•: : - : ; : 


25 


36 


":5ioi 


264 


892940 
892^39 


16S 


902160 


432 


::-:_: 


24 


3" 


79D259 


263 


:.: 


902420 


432 


: - i - : 


23 


38 


-:54I7 


203 


802-39 
892:!: 


10S 


902:": 


432 




22 


39 


795575 


2o3 


168 


902938 


432 


19706a 


21 


4o 


795-33 


20-3 


892536 


168 


903197 


43i 


do68o3 


20 


4i 


g.-CDSgi 


263 


2435 


169 


9-9o3456 


43 1 


10-096544 


\t 


42 


796049 


263 


892334 




903-14 


43 1 


0962S6 


43 


79620a 


263 


8c2233 


169 


90 1 : " : 


43 1 


096027 


1" 


44 


79*364 


262 


8 9 2l32 


169 


904232 


43 1 


095-68 


It 


45 


796521 


262 


892030 


169 


904491 


43i 


095509 


i5 


46 


796679 


262 


891929 
::::2- 


169 


904-5o 


43 1 


095250 


14 


4" 


79683C 


262 


IOC 


9c5oo8 


43 1 


- 


i3 


48 


- 5 : : : 5 


262 


891-26 


109 


905267 
905526 


_:: 


094733 


12 


4c 


— 1 5 : 


261 


891624 




43 1 




1 1 


5o 


797307 


261 


891023 


l _ : 


:::-:: 


43 1 


094210 


10 


5i 


! 9-797464 


261 


9-891421 


170 


9-906043 


43 1 


10-093957 

co. : :c5 


I 


02 


797621 


261 


891319 


1-0 


906302 


43 1 


53 




201 


891217 


1-0 


: : : ' : : 


43i 


99344c 


- 


54 


797934 


261 


891 1 i5 


1-0 


906819 


43 1 


093181 


: 


55 


798091 


261 


i 


1-0 


: : 


_. : : 


: : : ; 2 5 


5 


56 


798247 


261 


89091 1 


1-0 


::-;:: 


43 1 


co:::4 


- 


5- 


798403 


20-0 


1-0 


907594 


43 1 


:;:_:: 


3 


53 


79356o 


200 


390707 


170 


907853 


43 1 


092147 


2 


5 9 


798716 


200 


8906 d : 


I"0 


9081 1 i 


43o 


: : ~ : : 


1 


60 


79887a 


260 


890 5o3 


170 


908369 


_:- 


091631 


1 


Cosine. 


D. 


Sine. 


Dl 


Ootang 


D. 


Tar,. 


12 


VO 














61° 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC. 


67 


89° 














14 


r()° 

/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 





9.798872 


260 


9 • 89o5o3 


170 


9-908369 
908628 


43o 


I«k.09l63l 


60 


1 


799028 


260 


890400 


171 


43o 


091372 


a 


2 


799 l 8 4 


260 


890298 


171 


908886 


43o 


09 i 1 i 4 


3 


799339 
7994o5 


259 


890195 


171 


909144 


43o 


090856 


& 


4 


259 


890093 
889990 


Hi 


909402 


43o 


090598 


56 


5 


799651 


209 


171 


909660 


43o 


090340 


55 


6 


799806 


259 


889888 


Hi 


909918 


43o 


090082 


54 


I 


799962 


25g 


88 97 85 


171 


910177 


43o 


089823 


53 


8001 17 


25 9 


889682 


171 


910435 


43 


089565 


52 


9 


800272 


258 


889579 


Hi 


910693 


43 


089307 


5i 


10 


800427 


258 


889477 


HI 


910961 


43o 


089049 


5o 


ii 


9'8oo582 


258 


9.889374 


172 


9-911209 


43o 


10-088791 
088533 


% 


12 


800737 


258 


889271 


172 


91 1467 


43o 


i3 


800892 


258 


889168 ' 


172 


91 1725 


43o 


088275 


47 


i4 


801047 


258 


889064 


172 


91 1982 


43o 


088018 


46 


i5 


801201 


258 


888961 


172 


912240 


43o 


087760 


45 


16 


8oi356 


207 


888858 


172 


912498 
912706 


43o 


087502 


44 


17 


8oi5n 


257 


P88 7 55 


172 


43o 


087244 


43 


18 


80 1 665 


257 


48865 1 


172 


913014 


429 


086986 


42 


19 


801819 
801973 


25 7 


888548 


172 


913271 


429 


086729 


4i 


20 


257 


888444 


I 7 3 


913529 


429 


086471 


4o 


21 


9-802128 


257 


9-888341 


I 7 3 


9-913787 


429 


io-o862i3 


It 


22 


802282 


256 


888237 


I 7 3 


914044 


429 


o85 9 56 


23 


802436 


256 


888 1 34 


I 7 3 


914302 


429 


o856 9 8 


3 7 


24 


8o258o 
802743 


256 


888o3o 


I 7 3 


9i456o 


429 


o8544o 


36 


25 


256 


887926 
887822 


1 7 3 


914817 


429 


o85i83 


35 


26 


802897 
8o3ooo 


256 


l 7 3 


915075 


429 


084925 


34 


27 


256 


887718 


I 7 3 


9i5332 


429 


084668 


33 


28 


8o32o4 


256 


887614 


l 7 3 


915590 


429 


084410 


32 


29 


8o3357 


255 


887510 


l 7 3 


916847 


429 


o84i53 


3i 


3o 


8o35u 


255 


887406 


174 


916104 


429 


o838 9 6 


3o 


3i 


9-8o3664 


255 


9-887302 


174 


9-916362 


429 


io-o83638 


3 


32 


8o38i 7 


255 


887198 


174 


9 1 66 1 9 


429 


o8338i 


33 


803970 


255 


887093 


174 


916877 


429 


o83i23 


27 


34 


804123 


255 


886989 
886885 


174 


917134 


429 


082866 


26 


35 


804276 


254 


H4 


917391 


429 


082609 


25 


36 


804428 


254 


886780 


H4 


917648 


429 


082352 


24 


37 


8o458j 


254 


886676 


174 


917906 


429 


082094 


23 


38 


804734 


254 


8865 7 i 


174 


918163 


428 


o8i83 7 


22 


3 9 


804886 


234 


886466 


174 


918420 


428 


o8i58o 


21 


40 


800039 


254 


886362 


175 


918677 


428 


o8i323 


20 


4i 


9-805191 


254 


9.886257 


l 7 5 


9-918934 


428 


10-081066 


!i 


42 


8o5343 


253 


886i52 


i 7 5 


919191 


428 


080809 


43 


8o5495 


253 


886047 


i 7 5 


919448 


428 


o8o552 


17 


44 


8o5647 


253 


885 9 42 


i 7 5 


91970,5 


428 


080295 
o8oo38 


16 


45 


805799 
805901 


253 


885837 


175 


919962 


428 


i5 


46 


253 


885 7 32 


i 7 5 


920219 


428 


079781 
079524 


14 


47 


806 1 o3 


253 


885627 


i 7 5 


920476 


428 


i3 


48 


806254 


253 


885522 


i 7 5 


920733 


428 


079267 


12 


49 


806406 


252 


8854i6 


i 7 5 


920990 


428 


079010 


11 


5o 


8o655 7 


252 


8853i 1 


176 


921247 


428 


070753 


10 


5i 


9.806709 


252 


9-8852o5 


176 


9-92i5o3 


428 


10-078497 


I 


52 


806860 


252 


885ioo 


176 


921760 


428 


078240 


53 


80701 1 


252 


884994 


176 


922017 


428 


077983 


7 


54 


807163 


252 


884889 


176 


922274 


428 


077726 


6 


55 


807314 


252 


884783 


176 


92253o 


428 


077470 


5 


56 


807465 


25l 


884677 


176 


922787 


428 


077213 


4 


5 7 


807615 


25l 


884572 


176 


923o44 


428 


076956 


3 


58 


807766 


231 


884466 


176 


9233oo 


428 


076700 


2 


5 9 


807917 


25l 


88436o 


176 


923557 


427 


076443 


I 


60 


808067 


25l 


884254 


H7 


923814 


427 


076186 




1 


/ 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tanjr. 


12<. 


>° 














50° 



58 


LOGARITHMIC SIN T ES, 


TANGENTS, ETC 


Table 11. 


40° 














139° 


i 


Sine. 


D. 


Cosine. 


D. | 


Tang. 


D. 


Cotang. 


/ 





9-808067 
808218 


25l 


9.884254 


177 


9-9238l4 


427 


10-076186 


60 


i 


25l 


884148 


H7 


924070 


427 


075930 


& 


2 


8o8368 


25l 


884042 


177 


924327 

924583 


427 


075673 


3 


8o85ig 


25o 


883 9 36 


177 


427 


075417 


57 


4 


808669 


25o 


883829 
883723 


177 


924840 


427 


075l6o 


56 


5 


808819 


25o 


177 


925096 


427 


074904 


55 


6 


808969 


200 


883617 


177 


925352 


427 


074648 


54 


I 


8091 19 


200 


8835io 


177 


925609 


427 


07439I 
074l35 


53 


809269 


200 


883404 


177 


925865 


427 


52 


9 


809419 


249 


883297 


178 


926122 


427 


073878 


5i 


10 


809069 


249 


883i 9 i 


178 


926378 


427 


073622 


5o 


II 


9-809718 


249 


o-883o84 


I7 f 


9-926634 


427 


10-073366 


% 


12 


809868 


249 


882977 
882871 


l 'l 


926890 


427 


073IIO 


i3 


810017 


249 


178 


927147 
927403 


427 


072853 


47 


U 


810167 


248 


882764 


178 


427 


072597 


46 


i5 


8io3i6 


882657 


178 


927659 


427 


072341 


45 


16 


8io465 


248 


882000 


178 


927910 


427 


072085 


44 


17 


810614 


248 


882443 


178 


928171 


427 


071829 
07l573 


43 


18 


810763 


248 


882336 


179 


928427 


427 


42 


*9 


810912 


248 


882229 


H9 


928684 


427 


071316 


41 


20 


811061 


248 


882121 


179 


928940 


427 


071060 


4o 


21 


9-811210 


248 


9-882014 


179 


9.929196 


427 


10-070804 


ll 


22 


8u358 


247 


881907 


H9 


929452 


427 


070548 


23 


811007 


247 


881799 


179 


929708 


427 


070292 


37 


24 


8n655 


247 


881692 


179 


029964 


426 


070036 


36 


25 


81 1804 


247 


881 584 


179 


930220 


426 


069780 


35 


26 


81 1952 


247 


881477 


179 


93o475 


426 


069525 


34 


11 


812100 


247 


881309 


H9 


930731 


426 


069269 
069013 


33 


812248 


247 


881261 


180 


930987 


426 


32 


29 


8i23 9 6 


246 


881 1 53 


180 


93 1 243 


426 


068757 


3i 


3o 


812544 


246 


881046 


180 


931499 


426 


o685oi 


3o 


3i 


9-812692 


246 


9-880938 
88o83o 


180 


9.931755 


426 


10-068245 


3 


32 


812840 


246 


180 


932010 


426 


067990 
067734 


33 


812988 


246 


880722 


180 


932266 


426 


27 


34 


8i3i35 


246 


8806 1 3 


180 


932022 


426 


067478 


26 


35 


8i3283 


246 


88o5o5 


180 


932778 


426 


067222 


25 


36 


8i343o 


245 


88o3 9 7 
880289 


180 


933o33 


426 


066967 


24 


37 


813578 


245 


181 


933289 
933540 


426 


06671 1 


23 


38 


813725 


245 


880180 


181 


426 


066455 


22 


39 


813872 


245 


880072 


181 


933800 


426 


066200 


21 


40 


814019 


245 


879963 


181 


934o56 


426 


060944 


20 


4i 


9-814166 


245 


9.879855 


181 


9-9343ii 


426 


io-o6568o 
o65433 


12 


42 


8i43i3 


245 


879746 


181 


934067 


426 


43 


814460 


244 


879637 


181 


934822 


426 


065178 


17 


44 


814607 


244 


879029 


181 


930078 


426 


064922 


16 


45 


8i4753 


244 


879420 


181 


935333 


426 


064667 


i5 


46 


814900 


244 


8793 1 1 


181 


935589 


426 


06441 1 


14 


% 


8i5o46 


244 


879202 


182 


9 35844 


426 


0641 56 


i3 


8i5io3 
8i5339 


244 


879093 
878984 
878870 


18-2 


936100 


426 


063900 


12 


49 


244 


182 


9 36355 


426 


o63645 


11 


5o 


81 5485 


243 


i8j 


9366H 


426 


063389 


10 


5i 


9-8i563i 


243 


9.878766 


182 


9- 9 36866 


425 


io-o63i34 


I 


52 


815778 


243 


878656 


182 


937121 


420 


062879 
06262J 


53 


815924 


243 


878547 


182 


937377 


420 


1 


54 


816069 


243 


878438 


182 


9 37632 


425 


062368 


6 


55 


816213 


243 


878328 


182 


937887 


425 


o62ii3 


5 


DO 


8i636i 


243 


878219 


1 83 


9 38i42 


425 


06 1 858 


4 


57 


816507 


242 


878109 


1 83 


9 383o8 


425 


061602 


3 


58 


8i6652 


242 


877099 
877890 
877780 


1 83 


9 386o3 


425 


06 1 347 


2 


5 9 


816798 


242 


1 83 


938908 


4a5 


o6ioq2 


I 


60 


816943 


242 


1 83 


939163 


425 


060837 


I St 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


>° 












49° 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC. 59 


41° 














138° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 
60 


o 


9-816943 


242 


9.877780 


i83 


9.939163 


425 


iO'o6o837 


i 


817088 


242 


87767O 


1 83 


939418 


425 


o6o582 


a 


2 


817233 


242 


877560 


1 83 


939673 


425 


060327 


3 


817379 


242 


877450 


1 83 


939928 


425 


060072 


57 


4 


817524 


241 


877340 


1 83 


94oi83 


425 


059817 


56 


5 


817668 


241 


87723o 


184 


940439 


425 


059561 


55 


6 


817813 


241 


877120 


184 


940694 


425 


059306 


54 


i 


817958 


241 


8770IO 


184 


940949 


425 


o5oo5i 
058796 


53 


8i8io3 


24i 


876899 


184 


941204 


425 


52 


9 


818247 


241 


876780 


184 


941459 
94i7i3 


425 


058541 


5i 


10 


8i83g2 


24i 


876678 


184 


425 


058287 


5o 


ii 


9-8i8536 


240 


9.876568 


184 


9-941968 


425 


io-o58o32 


% 


12 


818681 


240 


876457 


184 


942223 


425 


o57777 


i3 


818825 


240 


876347 


184 


942478 


425 


057522 


47 


i4 


818969 
8191 i3 


240 


876236 


i85 


942733 


425 


057267 


46 


i5 


240 


876125 


i85 


942988 


425 


057012 


45 


16 


819207 


240 


876014 


1 85 


943243 


425 


056757 


44 


\l 


819401 


240 


875904 


1 85 


9434Q8 
943752 


425 


o565o2 


43 


819545 


239 


875793 
8 7 5682 


i85 


425 


o56248 


42 


19 


819689 


239 


1 85 


944007 


425 


055993 


4i 


20 


819832 


239 


875571 


i85 


944262 


425 


o55 7 38 


4o 


21 


9.819976 


239 


9-875459 
875348 


i85 


9-944517 


425 


10- 055483 


ll 


22 


820120 


239 


1 85 


944771 


424 


055229 


23 


820263 


239 


875237 


1 85 


945026 


424 


054974 


37 


24 


820406 


239 

238 


875126 


186 


945281 


424 


054719 
o54465 


36 


25 


82o55o 


875014 


186 


945535 


424 


35 


26 


820693 
82o836 


238 


874903 


186 


945790 


424 


o542io 


34 


i 


238 


874791 
874680 


186 


946045 


424 


o53955 


33 


820979 


238 


186 


946299 


424 


053701 


32 


29 


821122 


238 


874568 


186 


946554 


424 


o53446 


3i 


3o 


821265 


238 


874456 


186 


946808 


424 


o53i92 


3o 


3i 


9-821407 


238 


9-874344 


186 


9.947063 


424 


10-052937 


ll 


32 


82i55o 


238 


874232 


187 


9473i8 


424 


052682 


33 


821693 
82i835 


23 7 


874I2I 


187 


947572 


424 


052428 


27 


34 


2 3 7 


874009 


187 


947827 


424 


052173 


26 


35 


821977 


23 7 


8 7 38 9 6 


187 


948081 


424 


051919 
o5i665 


25 


36 


822120 


23 7 


873784 


187 


948335 


424 


24 


ll 


822262 


23 7 


8 7 36 7 2 


187 


948590 


424 


o5i4io 


23 


822404 


23 7 


873560 


187 


948844 


424 


o5n56 


22 


39 


822546 


23 7 


873448 


187 


949099 
9493D3 


424 


050901 


21 


40 


822688 


236 


873335 


187 


424 


050647 


20 


41 


9-82283o 


236 


9.873223 


187 


9 - 949608 


424 


io-o5o392 
o5oi38 


!§ 


42 


822972 


236 


873HO 


188 


949862 


424 


43 


823ii4 


236 


872998 
8 7 2&85 


188 


950116 


424 


049884 


\l 


44 


823255 


236 


188 


95o37 I 


424 


049629 
049373 


45 


823397 
823539 


236 


872772 
872659 


188 


95o625 


424 


i5 


46 


236 


188 


950879 
9§i i33 


424 


049121 


14 


% 


823680 


235 


872547 


188 


424 


048867 


i3 


823821 


235 


872434 


188 


95i388 


424 


048612 


12 


49 


823 9 63 


235 


872321 


188 


951642 


424 


048358 


11 


5o 


824104 


235 


872208 


188 


951896 


424 


048104 


10 


5i 


9.824245 


235 


9-872095 
871981 
871868 


189 


9'952i5o 


424 


10*047850 


I 


52 


824386 


235 


189 


. 952405 


424 


047595 


53 


824527 
824668 


235 


189 


952659 


424 


047341 


I 


54 


234 


871755 


189 


952913 


424 


047087 


55 


824808 


234 


871641 


189 


953167 


423 


046833 


5 


56 


824949 


234 


871528 


189 


953421 


423 


046579 
t)4632D 


4 


57 


825090 
82523o 


234 


87 1 41 4 


189 


953675 


423 


3 


58 


234 


871301 


189 


953929 
9 54 1 83 


423 


046071 


2 


59 


825371 


234 


871187 


189 


423 


045817 
045563 


1 


60 


8255n 


234 


871073 


190 


954437 


423 





1 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


1 


181 


s 












48° 



60 


LOGARITHMIC SINES, 


TANGENTS, ETC. Table IL 


42° 














137° 


i 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 





9-8255ii 


234 


9-871073 


190 


9-954437 


423 


10-045563 


60 


i 


8256'n 


233 


870960 
870846 


190 


954691 


423 


045309 


M 


2 


825791 


233 


190 


954946 


423 


o45o54 


3 


825931 


233 


870-32 


190 


955200 


423 


044800 


57 


4 


826071 


233 


870618 


T90 


955454 


423 


044546 


56 


5 


8262 1 1 


233 


870304 


190 


955708 


423 


044292 


55 


6 


82635i 


233 


870390 


190 


955961 


423 


044039 


54 


7 


826491 


233 


870276 


190 


9362l5 


423 


043783 


53 


8 


826631 


233 


87OIOI 


190 


•936469 


423 


04353 1 


52 


9 


826770 


232 


870047 


191 


936723 


423 


043277 


5i 


10 


826910 


232 


869933 


191 


936977 


423 


o43o23 


5o 


ii 


9-827049 


232 


9-869818 


191 


9-95723l 


423 


10-042769 


% 


12 


827189 


232 


869704 
869589 


191 


957485 


423 


042513 


i3 


827328 


232 


191 


957739 
957993 


423 


042261 


% 


14 


827467 


232 


869474 


191 


423 


042007 


i5 


827606 


232 


86g36o 


191 


958247 


423 


041753 


45 


16 


827745 


232 


869245 


191 


9585oo 


423 


041 5oo 


44 


17 


827884 


23l 


869130 


191 


9 58 7 54 


423 


041246 


43 


18 


828023 


23l 


869015 


192 


959008 


423 


040992 
040738 


42 


!9 


828162 


23l 


8689OO 


192 


939262 


423 


4i 


20 


8283oi 


23l 


868783 


192 


939516 


423 


040484 


4o 


21 


9-828439 
828578 


23l 


9-868670 


192 


9.959769 
960023 


423 


io-o4o23i 


M 


22 


23l 


868555 


I92 


423 


039977 


23 


828716 


23l 


868440 


192 


960277 


423 


039723 


37 


24 


828855 


23o 


868324 


192 


96o53o 


423 


039470 


36 


25 


828993 
829131 


23o 


868209 


192 


960784 


423 


039216 
038962 


35 


26 


23o 


868o 9 3 


192 


961038 


423 


34 


27 


829269 


23o 


867978 


193 


961292 


423 


038708 


33 


28 


829407 


23o 


867862 


193 


961543 


423 


o38455 


32 


2 9 


829543 


23o 


867747 


193 


961799 


423 


o382oi 


3i 


3o 


829683 


23o 


867631 


193 


962052 


423 


037948 


3o 


3i 


9-829821 


229 


9-8675i5 


193 


9.962306 


423 


10-037694 


M 


32 


829959 


229 


867399 
867283 


193 


962360 


423 


037440 


33 


830097 


229 


193 


962813 


423 


037187 


27 


34 


830234 


229 


867167 


193 


963067 


423 


036933 


26 


35 


83o372 


229 


867051 


193 


963320 


423 


o3668o 


23 


36 


83o5o9 


229 


866935 
866819 
866703 


194 


963574 


423 


o36426 


24 


ll 


83o646 


229 


194 


963828 


423 


036172 


23 


830784 


229 


194 


964081 


423 


035919 


22 


3 9 


830921 


228 


866586 


194 


964335 


423 


o35663 


21 


4o 


83io58 


228 


866470 


I 9 4 


964588 


422 


o354i2 


20 


4i 


o-83i io5 

83i332 


228 


9-866353 


194 


9.964842 


422 


io-o35i58 


\% 


42 


228 


866237 


194 


965095 


422 


o349o5 


43 


831469 


228 


866120 


194 


965349 


422 


o3465i 


17 


44 


83 1606 


228 


866004 


193 


965602 


422 


034398 


16 


45 


83i 7 42 


228 


865887 


195 


965855 


422 


o34U5 


i5 


46 


831879 


228 


865770 


193 


966109 


422 


o338 9 i 


14 


47 


832013 


227 


865653 


ig5 


966362 


422 


o33638 


i3 


48 


832ID2 


227 


865536 


195 


966616 


422 


o33384 


12 


49 


832288 


227 


865419 


193 


966869 


422 


o33i3i 


11 


5o 


832425 


227 


8653oa 


193 


967123 


422 


032877 


10 


5i 


o-83256i 


227 


9-865i85 


i 9 5 


9-967376 


422 


10-032624 


\ 


52 


832697 
832833 


227 


865o68 


i 9 5 


967629 


422 


032371 


53 


227 


864950 


195 


967883 


422 


032117 


I 


54 


832969 


226 


864833 


196 


968 1 36 


422 


o3i864 


55 


833 1 od 


226 


864716 


196 


9 6838 9 
968643 
968896 


422 


o3i6u 


5 


56 


833241 


226 


864398 
864481 


196 


422 


o3 1 337 


4 


^>7 


8333 77 


226 


196 


422 


o3iic4 


3 


58 


8335i2 


226 


864363 


196 


969149 


422 


o3oS5i 


2 


5 9 


833648 


226 


864245 


196 


969403 


422 


030397 


1 


60 


833 7 83 


226 


864127 


106 


969656 


422 


o3o344 





t 


Cosine. 


D. 


Sine. 


D. 


Cotang. 


a 


Tang. 


\%\ 















470 



Table II. LOGARITHMIC SINES, 


TANGENTS, ETC. 


61 


43° 














136° 


/ 


Sine. 


D. 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


t 





9-833783 


226 


9.864127 


196 


9-969656 


422 


io«o3o344 


60 


i 


833919 


225 


864010 


196 


969909 


422 


030091 
029838 


n 


2 


834t>54 


225 


8638 9 2 


197 


970162 


422 


3 


834189 


225 


863774 


197 


970416 


422 


029584 


57 


4 


834325 


225 


863656 


197 


970669 


422 


02933l 


56 


5 


83446o 


225 


863538 


197 


970922 


422 


029078 


55 


6 


834595 
834730 


225 


863419 


197 


971175 


422 


028825 


54 


I 


225 


8633oi 


197 


97U29 


422 


028571 


53 


834865 


225 


863 1 83 


197 


971682 


422 


0283 18 


52 


9 


834999 


224 


863o64 


197 


971935 


422 


028o65 


5i 


10 


835i34 


224 


862946 


198 


972188 


422 


027812 


5o 


U 


9.835269 
8354o3 


224 


9.862827 


198 


9-972441 


422 


10-027559 
0273oD 


% 


12 


224 


862709 


198 


972695 


422 


i3 


835538 


224 


862590 


198 


972948 


422 


027052 


47 


14 


8356 7 2 


224 


862471 


198 


973201 


422 


026799 


46 


i5 


835807 


224 


862353 


198 


973454 


422 


026546 


45 


16 


835941 


224 


862234 


198 


973707 


422 


026293 


44 


\l 


836075 


223 


8621 i5 


198 


973960 


422 


026040 


43 


836209 
836343 


223 


861996 

861877 


198 


974213 


422 


025787 


42 


19 


223 


198 


974466 


422 


025534 


4i 


20 


836477 


223 


861758 


199 


974720 


422 


020280 


40 


21 


9-8366ii 


223 


9-86i638 


199 


9-974973 


422 


10-025027 


ll 


22 


836 7 45 
836878 


223 


86i5i 9 


199 


975226 


422 


024774 


23 


223 


861400 


199 


975479 


422 


024521 


37 


24 


837012 


222 


861280 


199 


97 5 7 32 


422 


024268 


36 


25 


837146 


222 


861161 


199 


975985 


422 


0240 1 5 


35 


26 


837279 


222 


861041 


199 


976238 


422 


023762 


34 


2 


837412 


222 


860022 


199 


976491 


422 


023509 


33 


837546 


222 


860802 


199 


976744 


422 


023256 


32 


29 


837679 ' 


222 


860682 


200 


976997 


422 


O23oo3 


3i 


3o 


837812 


222 


86o562 


200 


977250 


422 


022750 


3o 


3i 


9.837945 


222 


9-860442 


200 


9-9775o3 


422 


10.022497 


3 


32 


838078 


221 


86o322 


200 


977756 


422 


022244 


33 


8382 1 1 


22[ 


860202 


200 


978009 


422 


021991 

021738 


27 


34 


838344 


221 


860082 


200 


978262 


422 


26 


35 


838477 


221 


859962 
85 9 842 


200 


97 85i5 


422 


021485 


25 


36 


8386io 


221 


200 


978768 


422 


021232 


24 


iz 


838742 


221 


859721 


201 


979021 


422 


020979 


23 


838875 


221 


859601 


201 


979274 


422 


020726 


22 


39 


839007 


221 


85 9 48o 


201 


979527 


422 


020473 


21 


40 


839140 


220 


85 9 36o 


201 


979780 


422 


O2022O 


20 


41 


9.839272 


220 


9-859239 


201 


9-980033 


422 


IO.OI9967 


\l 


42 


839404 


220 


85 9 i 19 

858 99 § 


201 


980286 


422 


OI97U 


43 


83 9 536 


220 


201 


98o538 


422 


OI9462 


\l 


44 


83 9 668 


220 


858877 


201 


980791 


421 


OI0209 


45 


83 9 8oo 


220 


858756 


202 


981044 


421 


018956 


i5 


46 


83 99 32 


220 


858635 


202 


981297 
9 8i55o 


421 


018703 


14 


% 


840064 


219 


8585i4 


202 


421 


Ol845o 


i3 


840196 


219 


8583 9 3 


202 


9 8i8o3 


421 


018197 


12 


49 


840328 


219 


8582 7 2 


202 


982056 


421 


017944 


11 


5o 


840459 


219 


858i5i 


202 


982309 


421 


OI769I 


10 


5i 


9-840591 


219 


9 -858020 


202 


9-982562 


421 


10-017438 





52 


840722 
84o854 


219 


857908 


2C2 


982814 


421 


OI7186 


8 


53 


219 


857786 


202 


983067 


421 


Ol6933 


I 


54 


840985 


219 


85 7 665 


203 


983320 


421 


Ol668o 


55 


841116 


2l8 


857543 


203 


983573 


421 


016427 


5 


56 


841247 
841378 


2l8 


85 7 422 


203 


983826 


421 


O16174 


4 


57 


2l8 


857300 


203 


984079 
984332 


421 


OI592I 


3 


58 


841509 


2l8 


85 7 , 7 8 


203 


421 


oi5668 


2 


5 9 


841640 


218 


85 7 o56 


203 


984584 


421 


oi54i6 


1 


60 


841771 


2l8 


856 9 34 


203 


984837 


421 


0i5i63 





/ 

13? 


Cosine. 


D. 


Sine. 


D 


Cotang. 


D. 


Tang. 


t 

















*6° 



62 


LOGARITHMIC SINES, 


TANGENTS, ETC 


Table II. 


44° 














185° 


t 


Sine. 


B. | 


Cosine. 


D. 


Tang. 


D. 


Cotang. 


/ 





9-84I77I 


218 


9«856934 


203 


9.984837 


421 


io«oi5i63 


60 


i 


841902 


218 


8568i2 


203 


985090 


421 


01 49 10 


is 


2 


842033 


218 


8566 9 o 


204 


985343 


421 


014657 


3 


842163 


217 


856568 


204 


985596 


421 


014404 


57 


4 


842294 


217 


856446 


204 


9 85848 


421 


oi4i52 


56 


5 


842424 


217 


856323 


204 


986101 


421 


013899 


55 


6 


842555 


217 


856201 


204 


986354 


421 


01 3646 


54 


I 


842685 


217 


806078 


204 


986607 


421 


oi3393 


53 


8428i5 


217 


855q56 

855833 


204 


986860 


421 


oi3i4o 


52 


9 


842946 


217 


204 


9871 12 


421 


012888 


5i 


10 


843076 


217 


85571 1 


205 


987365 


421 


012635 


5o 


il 


9.843206 


2l6 


9-855588 


205 


9.987618 


421 


IO.OI2382 


% 


12 


843336 


2l6 


855465 


205 


987871 


421 


012129 


i3 


843466 


2l6 


855342 


205 


988123 


421 


01 1877 


% 


14 


843595 


2l6 


855219 


205 


988376 


421 


01 1624 


i5 


843725 


2l6 


855096 


205 


988629 


421 


011371 


45 


16 


843855 


2l6 


854973 
85485o 


205 


988882 


421 


011118 


44 


*7 


843984 


2l6 


205 


Q89134 


421 


010866 


43 


18 


8441 14 


2l5 


854727 


206 


989387 


421 


oio6i3 


42 


'9 


844243 


2l5 


8546o3 


206 


989640 


421 


oio36o 


4i 


20 


844373 


2l5 


85448o 


206 


989893 


421 


010107 


4o 


21 


9- 8445o2 


2l5 


9-854356 


206 


9.990145 


421 


10-009835 


1 


22 


844631 


2l5 


854233 


206 


990398 


421 


009602 


23 


844760 


2l5 


854109 


206 


990631 


421 


009349 


37 


24 


844889 
845oi8 


2l5 


853 9 86 


206 


990903 


421 


009097 
008844 


36 


25 


2l5 


853862 


206 


991 i 56 


421 


35 


26 


845147 


2l5 


853 7 38 


206 


991409 


421 


008591 


34 


il 


845276 


214 


853614 


207 


991662 


421 


oo8338 


33 


8454o5 


214 


853490 


207 


991914 


421 


008086 


32 


29 


845533 


214 


853366 


207 


992167 


421 


007833 


3i 


3o 


845662 


214 


853242 


207 


992420 


421 


007680 


3o 


3i 


9-845790 


214 


9-853n8 


207 


9.992672 


421 


10-007328 


11 


32 


845919 


214 


852994 
852869 


207 


992925 


421 


007075 


33 


846047 


214 


207 


993178 
99343i 


421 


006822 


27 


34 


846175 


214 


852743 


207 


421 


006669 


26 


35 


8463o4 


214 


852620 


207 


993683 


421 


006317 


25 


36 


846432 


213 


852496 


208 


993936 


421 


006064 


24 


37 


84656o 


213 


852371 


208 


994189 


421 


oo58u 


23 


38 


846688 


213 


852247 


208 


994441 


421 


oo5559 


22 


39 


846816 


213 


852122 


208 


994694 


421 


oo53o6 


21 


40 


846944 


213 


85 1 997 


208 


994947 


421 


oo5o53 


20 


4i 


9-847071 


2l3 


9.851872 


208 


9.995199 


421 


10-004801 


is 


42 


847199 


213 


861747 


208 


995432 


421 


004348 


43 


847227 


213 


85i622 


208 


996703 


421 


004295 


17 


44 


847454 


212 


85i497 


209 


995937 


421 


oo4o43 


16 


45 


847582 


212 


85i3 7 2 


209 


996210 


421 


003790 


i5 


46 


847709 
847836 


212 


85 1 246 


209 


996463 


421 


oo3537 


14 


a 


212 


85ii2i 


209 


9967 1 5 


421 


oo3285 


i3 


847964 


212 


860996 


209 


996968 


421 


oo3o32 


12 


49 


848091 


212 


860870 


209 


997221 


421 


002779 


11 


5o 


848218 


212 


85o745 


209 


997473 


421 


002327 


10 


5i 


9-848345 


212 


9-85o6io 
85o49J 


209 


9.997726 


421 


IO-002274 


I 


5a 


848472 


211 


210 


997979 


421 


O0202I 


53 


848599 


211 


85o368 


210 


998231 


421 


OOI769 

ooi5i6 


I 


54 


848726 


211 


85 242 


210 


998484 


421 


55 


848852 


211 


85oii6 


210 


998737 


421 


001263 


5 


56 


848979 


211 


849990 


210 


998989 


421 


OOIOII 


4 


57 


849106 


211 


849864 


210 


999242 


421 


000758 


3 


58 


849232 


211 


849738 


210 


999495 


421 


ooo5o5 


2 


59 


849359 


211 


8496 1 1 


2IO 


999747 


421 


ooo253 


1 


6c 

/ 


849483 


211 


84g485 


210 


I0< 000000 


421 


io> 000000 





Cosine. 


D. 


Sine. 


D. 


Cotang. 


D. 


Tang. 


13 


1° 












45° 



T A B L E III., 

OF 

NATURAL SINES AND TANGENTS', 

TO 
EVERY DEGREE AND MINUTE OF THE QUADRANT. 



If the given angle is less than 45°, look for the degrees and the title of the 
column, at the top of the page ; and for the minutes on the left. But if the angle 
ib between 45° and 90 , look for the degrees and the title of the column, at the 
bottom ; and for the minutes on the right. 

The Secants and Cosecants, which are not inserted in this table, may be easily 
supplied. If 1 be divided by the cosine of an arc, the quotient will be the secant 
of that arc. And if 1 be divided by the sine, the quotient will be the cosecant. 

The values of the Sines and Cosines are less than a unit, and are given in deci- 
mals, although the decimal point is not printed. So also, the tangents of arcs lets 
than 45°, and cotangents of arcs greater than 45°, are less thar a unit tml ore ex- 
pressed in decimals with the decimal point omitted. 



64 NATURAL SINES AND COSINES. Table III. 


/ 


0° 


1° 


2° 


r 
3° 


40 


/ 


Sine. 


Cosine. 


Sine. Cosine. 


Sine. Cosine. 


Sine. 1 Cosine. 


Sine. Cosine. 





00000 


Unit. 


01745 99985 


o3490 


99939 


o5234 99863 


06976 1 99756 


60 


i 


00029 
ooo58 


Unit. 


01774 99984 


o35l9 


99938 


o5263 99861 


07005 


99754 


51 


2 


Unit. 


oi8o3 99984 


03548 


99937 


05292 


99860 


07034 


99752 


3 


00087 


Unit, 


0i832 99983 


03577 
o36o6 


99936 


o532i 


99808 


07063 


99750 


57 


4 


00116 


Unit. 


01862 99983 


99935 


o535o 


99 85 7 


07092 


99748 


56 


5 


ooi45 


Unit. 


01891 1 99982 


o3635 


99934 


05379 
o54o8 


99855 


07121 


99746 


55 


6 


00175 


Unit. 


01920 | 99982 


o3664 


99933 


99854 


07150 


99744 


54 


7 


00204 


Unit. 


01949 999 Sl 


03693 


999 32 


o5437 


99852 


07170 


99742 


53 


8 


00233 


Unit. 


01978 


99980 


o3723 


9993i 


o5466 


99851 


07208 


99740 


52 


9 


00262 


Unit, 


02007 


99980 


03752 


99930 


o5495 


99849 


07237 ( 99738 
07266 99736 


5i 


10 


00291 


Unit. 


02o36 


99979 


03781 


99929 


o5524 


99847 


5o 


ii 


00320 


99999 


o2o65 


99979 
99978 


o38io 


99927 


o5553 


99846 


07295 99734 


49 


12 


oo34o 


99999 


02094 


o3839 


99926 


o5582 


99844 


07324 9973 1 


48 


i3 


00378 


99999 


02123 


99977 


o3868 


99925 


o56u 


99842 


07353 


99729 


47 


14 


00407 


99999 


02l52 


99977 


03897 


99924 


o564o 


99841 


07382 


99727 


46 


i5 


oo436 


99999 


02l8l 


99976 


03926 


99923 


05669 


99839 


0741 1 


99725 


45 


16 


oo465 


99999 


022II 


99976 


o3955 


99922 


05698 


9 9 838 


07440 


99723 


44 


H 


00495 99999 


02240 


99973 


03984 


99921 


05727 


99836 


07460 
07498 


99721 


43 


18 


oo524 


99999 


O2269 


99974 


o4oi3 


99919 
99918 


05736 


99 834 


99719 


42 


19 


oo553 


99993 


02298 


99974 


04042 


o5785 


99 833 


07527 


99716 


4i 


20 


oo582 


99998 


02327 


99973 


04071 


99917 


o58i4 


9983 1 


07556 


99714 


4o 


21 


0061 1 


99998 


02356 


99972 


04100 


99916 


o5844 


99829 


07585 


99712 


3 Q 


22 


00640 


99998 


02385 


99972 


04129 


99913 


o58 7 3 


99827 


07614 


99710 


38 


23 


00669 
00698 


99998 


02414 


99971 


041 5o 


99913 


05902 


99826 


07643 


99708 


5 7 


24 


99998 


02443 


99970 


04188 


99912 


o593i 


99824 


07672 


99703 


36 


25 


00727 


99997 


02472 


99969 


04217 


999 1 1 


05960 


99822 


07701 


997 o3 


35 


26 


00756 


99997 


02301 


99960 


04246 


99910 


05980 


99821 


07730 


99701 


34 


3 


00785 


99997 


02530 


99968 


04275 


99909 


06018 


99819 


07759 


99699 


33 


00814 


99997 


0256o 


99967 


o43 04 


99907 


06047 


99817 


07788 
07817 


99696 


32 


29 


00844, 


99996 


02589 j 99966 


04333 


99906 


06076 


99815 


99694 


3i 


3o 


00873 


99996 


02618 1 99966 


04362 


99905 


06103 


99813 


07S46 


99692 


3o 


3i 


00902 


99996 


0264* 99965 


04391 


99904 


o6i34 


99812 


07873 


99689 


8 


32 


00931 


99996 


02670 j 99964 


04420 


99902 


06 1 63 


99810 


07904 


99687 


33 


00960 


99993 


02705 


99963 


o444o 
04478 


99901 


06192 


99808 


07933 


99685 


27 


34 


00989 
01018 


99993 


02734 


99963 


99900 
99898 


06221 


99806 


07962 


99683 


26 


35 


99993 


02763 


99962 


04507 


o625o 


99804 


07991 


99680 


25 


36 


01047 


99993 


02792 


99961 


04536 


99897 


06279 
o63o8 


99803 


08020 


99678 


24 


u 


01076 


99994 


02821 


99960 


04565 


99896 


99801 


08049 


99676 


23 


ouo5 


99994 


o2S5o 


99939 


04594 


99894 


o6337 


99799 


08070 


99673 


22 


39 


on34 


99994 


02S79 
02908 


99939 
99938 


04623 


99893 


o6366 


99797 


08107 


99671 


21 


40 


01164 


99993 


04653 


99892 


06395 


99793 


o8i36 


99668 


20 


41 


01193 


99993 


02938 


99937 


04682 


99800 
99889 
99888 


06424 


9979 3 


o8i65 


99666 


:? 


42 


01222 


99993 


02967 


99956 


047 1 1 


o6453 


99792 


08194 


99664 


43 


OI25l 


99992 


02996 


99955 


04740 


06482 


99700 


0S223 


99661 


17 


44 


01280 


99992 


o3o25 


99954 


04760 
04798 


99886 


o65u 


99788 


08232 


99 65 9 


16 


45 


01309 


99991 


o3o54 


99933 


998S5 


o654o 


997S6 


08281 


99637 


i5 


46 


oi338 


99991 


o3o83 


999 52 


04827 


99883 


06569 


99784 


o83io 


99634 


14 


s 


01367 


99991 


03lI2 


99932 


04856 


99S82 


06598 


99782 


oS33 9 
o8368 


99652 


i3 


01396 


99990 


o3i4i 


99931 


04885 


99881 


06627 


9978o 


99649 


12 


49 


01425 


99990 
999*9 


03170 


99930 


04914 


99S79 


o6656 


99778 


08397 


99647 


11 


30 


oi454 


03199 

03228 


99949 


04943 


99878 


066 8 5 


99776 


0S426 


99644 


10 


5i 


01483 


99989 


99948 


04972 


99876 


06714 


99774 


oS455 


99642 


I 


52 


oi5i3 


99989 
99988 


o3257 


999^7 


o5ooi 


99875 


06743 


99772 


0S484 


99639 


53 


oi542 


o32S6 


99946 


o5o3o 


99873 


06773 


99770 


o85i3 


99637 


I 


54 


0071 


99988 


o33i6 


99943 


o5o5o 


99872 


06802 


99768 


0S342 


99635 


55 


01600 


99987 


o3345 


99944 


o5o88 


99870 


o683i 


99766 


08571 


99632 


5 


56 


01629 
oi658 


99987 


o3374 


99943 


o5ii7 


99869 


06860 


99"64 


0S600 


99630 


4 


57 


99986 


o34o3 


99942 


o3i46 


99867 


06889 


99762 


08629 
o8658 


99627 


3 


58 


01687 


99986 


o3432 


99941 


o5i75 


99866 


06918 


99760 


99623 


2 


5 9 


01716 


99 9 85 


o346i 


99940 


03205 


99S64 


06947 


997 58 


0S6S7 


99622 


1 


00 


01745 


99983 


03490 


99939 


o5234 


99S63 


06976 


99756 


0S716 


99619 







Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine, j Sine. 




' 


89° 


88° 


S7° 


S6> 


85° 





Table III. 


NATURAL SINES AND COSINES. 


65 




5° 


6° 


7° 


8° 


9° 


. 




| Sine 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. Cosine. 


60 


o 


08716 


99619 


10453 


99452 


12187 


99255 


i3 9 . 7 


99027 


1 5643 


98769 


i 


08745 


99617 


10482 


99449 


I22l6 


9925i 


13946 


99023 


15672 


98764 


n 


2 


08774 


99614 


io5u 


99446 


12245 


99248 


i3 97 5 


99019 
990 1 5 


15701 


98760 


3 


o88o3 


99612 


io54o 


99443 


12274 


99244 


1 4004 


1 5730 


98755 


57 


4 


o883i 


99609 


10569 


99440 


12302 


99240 


i4o33 


9901 1 


i5 7 58 


9 8 7 5i 


56 


5 


08860 


99607 


10597 


99437 


i233i 


99237 


1 406 1 


99006 


i5 7 8 7 


98746 


55 


6 


08889 
08918 


99604 


10626 


99434 


i236o 


99233 


14090 


99002 
98998 


i58i6 


98741 


54 


7 


99602 


io655 


9943l 


I238 9 


99230 


14119 

14148 


1 5845 


98787 


53 


8 


08947 


99599 


10684 


99428 


12418 


99226 


98994 


i58 7 3 


98732 


52 


9 


08976 


99596 


10713 


99424 


12447 


99222 


14177 
i42o5 


98990 


1 5902 


98728 


5i 


10 


09005 


9 9 5 9 4 


10742 


99421 


12476 


99219 
992 1 5 


98986 


l l 9 l X 


98723 


5o 


ii 


09034 


9 9 5oi 
99588 


10771 


994l8 


i25o4 


14234 


98982 


15959 

1 5 9 88 


98718 


t 


12 


09063 


10800 


994 1 5 


12533 


992 1 1 


14263 


98978 


987U 


i3 


09092 


99086 


10829 


99412 


12562 


99208 


14292 


98973 


16017 


98709 


47 


i4 


091 21 


99583 


io858 


99409 


1 2591 


99204 


14320 


98969 


16046 


98704 


46 


i5 


09150 


99580 


10887 


99406 


12620 


99200 


14349 


98965 


16074 


98700 


45 


16 


09170 


99578 


10916 


99402 


12649 


99197 


i43 7 8 


98961 


i6io3 


98695 


44 


n 


09208 


99573 


10945 


99399 


12678 


99193 
99189 


14407 


98957 


i6i32 


98690 


43 


18 


09237 


99572 


10973 


99396 


12706 


14436 


98953 


1 61 60 


98686 


42 


'9 


09266 


99370 


1 1002 


99393 


12735 


99186 


14464 


98948 


16189 

16218 


98681 


41 


20 


09295 


99367 


no3i 


99390 


12764 


99182 


14493 


98944 


98676 


40 


21 


09324 


99564 


1 1 060 


99386 


12793 


99178 


14522 


98940 


16246 


98671 


i* 


22 


09353 


99562 


1 1089 


99 383 


12822 


99175 


i455i 


9 8 9 36 


16275 


98667 


23 


09382 


99559 


11118 


99380 


12801 


99171 


U58o 


98931 


1 63 04 


98662 


37 


24 


0941 1 


99556 


1 1147 


99 3 77 


12880 


99167 


14608 


98927 


16333 


98657 


36 


25 


09440 


99553 


11176 


99374 


12908 


99163 


I463 7 


98923 


i636i 


98652 


35 


26 


09469 


9955i 


II205 


99370 


12937 


99160 


14666 


98919 


16390 


98648 


34 


11 


09498 


99348 


1 1 234 


99367 


12966 


99156 


14695 


98914 


16419 


98643 


33 


09527 


99545 


1 1 263 


99364 


12995 


99152 


14723 


98910 


16447 


9 8638 


32 


29 


09306 


99542 


11291 


99360 


i3o24 


99148 


14752 


98906 


16476 


98633 


3i 


3o 


09585 


99540 


Il320 


99357 


i3o53 


99144 


14781 


98902 


i65o5 


98629 


3o 


3i 


09614 


99537 


1 1 349 


99354 


i3o8i 


99' 4i 


14810 


98897 


1 6533 


98624 


29 


32 


09642 


99534 


11378 


993 5 1 


i3i 10 


99137 


14838 


98893 


i6562 


98619 


28 


33 


09671 


9953i 


1 1407 


99347 


i3i3 9 


99i33 


14867 


98889 


i65 9 i 


98614 


27 


34 


09700 


99528 


1 1436 


99344 


i3i68 


99129 


14896 


98884 


16620 


98609 


26 


35 


09729 


99526 


1 1465 


99341 


i3i 97 


99 1 2D 


14925 


98880 


16648 


98604 


25 


36 


09758 


99523 


1 1 494 


99337 


13226 


99122 


14954 


98876 


16677 


98600 


24 


37 


09787 


99520 


u523 


99334 


i3254 


991 18 


14982 


98871 


16706 


9 85 9 5 


23 


38 


09816 


99 5i 7 


n552 


9933i 


i3 2 83 


99 ,J 4 


i5oh 


98867 


16734 


98590 


22 


3 9 


09845 


995i4 


u58o 


99327 


i33ia 


991 10 


i5o4o 


9 8863 


i6 7 63 


9 8585 


21 


4o 


09874 


995i 1 


1 1 609 
n638 


69324 


i334i 


99106 


15069 


9 8858 


16792 
16820 


9 858o 


20 


4i 


09903 


99508 


99320 


13370 


99102 


i5o 9 7 


9 8854 


9 85 7 5 


\l 


42 


09932 


99506 


1 1 667 


99317 


13399 


99098 


i5i26 


98849 
98843 


16849 
16878 


98570 


43 


09961 


995o3 


11696 


993i4 


13427 


99094 


i5i55 


98565 


17 


44 


09990 


995oo 


11725 


99310 


1 3456 


99001 


i5i84 


98841 


16906 


9856i 


16 


45 


10019 


99497 


.11754 


99307 


1 3485 


99087 


15212 


9 8836 


i6 9 35 


9 8556 


i5 


46 


10048 


99494 


1 1783 


993o3 


i35i4 


99083 


i524i 


98832 


16964 


9855i 


14 


47 


10077 


9949 > 


11812 


993oo 


i3543 


99079 
99075 


15270 


98827 


16992 


98546 


i3 


48 


10106 


99488 


1 1 840 


99297 


13572 


15209 


98823 


1 702 1 


98541 


12 


49 


ioi35 


99485 


1 1869 


99293 


i36oo 


99071 


15327 


98818 


17050 


9 8536 


1 2 


5o 


1 01 64 


99482 


,.898 


99200 

99286 


13629 


99067 


1 5356 


98814 


17078 


9 853 1 


10 


5i 


10192 


99479 


1 1927 


i3658 


99063 


15385 


98809 


17107 


98526 


9 


52 


10221 


99476 


1 1956 


99283 


13687 


99o5o 
99055 


i54U 


98803 


17136 


98521 


8 


53 


10250 


99473 


1 1985 


99279 


i3 7 i6 


1 5442 


98800 


17164 


9 85 1 6 


7 


54 


10279 


99470 


12014 


99276 


i3 7 44 


9905 1 


1 547 1 


98796 


17193 


9 85u 


6 


55 


io3o8 


99467 


12043 


99272 


i3 7 73 
i38o2 


99047 


i55oo 


98791 
98787 


17222 


9 85o6 


5 


56 


io337 


99464 


1 207 1 


99269 
99265 


99043 


i5529 


i725o 


985oi 


4 


u 


io366 


9946i 


1 2 100 


i383i 


99039 
99035 


1 555 7 


98782 


17308 


98496 


3 


10395 


99458 


12129 

I2i58 


99262 


1 386o 


15586 


98778 


98491 


2 


5o 


10424 99455 


99258 


i388 9 


9903 1 


i56i5 


98773 


17336 


98486 


1 


60 


io453 1 99452 


12187 


99255 


i3 9 i 7 


99027 


1 5643 


98769 


17365 


98481 




> 


f 


Cosine. 


Sine. 


vbsine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sin«. 


Cosine. 


Sine. 


84° 


83° 


82° 


81° 


80° 



41 



G6 


NATURAL SINES AND COSINES. Table III 


r 


10° 


11° 


12° 


13° 


14° 


1 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. | 


Cosine. 





17365 


98481 


1 908 1 


98163 


20791 


97815 


22495 


97437 


24192 


97o3o 


60 


i 


17393 


98476 


191OQ 


9 8;5 7 


20820 


97809 


22523 


9743o 


24220 


97023 


58 


2 


17422 


98471 


I 9 l38 


98l52 


20848 


97803 


22552 


97424 


24249 


97015 


3 


I745l 


98466 


I9167 


98146 


20877 


97797 


22580 


97417 


24277 


97008 


57 


4 


17479 


98461 


i 9I9 5 


98140 


20905 


9779 1 


226o8 


9741 1 


243o5 


97001 


56 


5 


17508 


98455 


19224 


98l35 


20933 


97784 


22637 


97404 


24333 


96994 


55 


6 


17537 


9845o 


19252 


98129 


20962 


97778 


22665 


97398 


24362 


96987 


54 


7 


17565 


98445 


19281 


98124 


20990 


97772 


22693 


97391 


243oo 


96980 


53 


8 


17094 


98440 


19309 


98118 


21019 


97766 


22722 


97384 


24418 


96973 


52 


9 


17623 


98435 


i 9 338 


98112 


21047 


97760 


22750 


97378 


24446 


96966 


5i 


10 


i 7 65i 


98430 


i 9 366 


98107 


21076 


97754 


22778 


97371 


24474 


96959 


5o 


ii 


17680 


98425 


19395 


98101 


21 104 


97748 


22807 


97365 


245o3 


96952 


% 


12 


17708 


98420 


19423 


98096 


2Il32 


97742 


22835 


97358 


2453 1 


96945 


i3 


17737 


984U 


19452 


9809O 


21l6l 


97735 


22863 


9735i 


24559 


969^ 


47 


i4 


17766 


98409 


1 948 1 


98084 


2Il8o 


97729 
97723 


22892 


97345 


24587 


96930 


46 


i5 


17794 


98404 


19509 


98079 


2I2I0 


22920 


97338 


246i5 


96923 


45 


r6 


17823 


98399 


i 9 538 


98073 


21246 


97717 


22948 


9733i 


24644 


96916 


44 


17 


17852 


9 83 9 4 


19566 


98067 


21275 


977H 


22977 


97325 


24672 


96909 


43 


18 


17880 


9 838 9 
98383 


ig5 9 5 


98061 


2i3o3 


97705 


23oo5 


973i8 


24700 


96902 
96894 
96887 


42 


19 


17909 


19623 


9 8o56 


2i33i 


97698 


23o33 


973n 


24728 


41 


20 


17937 


98378 


19652 


98o5o 


2i36o 


97692 


23o62 


973o4 


24756 


40 


21 


17966 


98373 


19680 


98044 


2 1 388 


97686 


23090 


97298 


24784 


96880 


ll 


22 


17995 


9 8368 


19709 


98039 


21417 


97680 


23n8 


97291 
97284 


248i3 


96873 


23 


18023 


9 8362 


19737 


9 8o33 


21445 


97673 


23 146 


24841 


96866 


37 


24 


i8o52 


98357 


19706 


98027 


21474 


97667 


23i75 


97278 


24869 


96858 


36 


25 


18081 


98352 


>9794 


98021 


2l502 


97661 


23203 


97271 


24897 


9685i 


35 


26 


18100 


98347 


19823 


98016 


2i53o 


97655 


2323l 


97264 


24925 


96844 


34 


3 


i8i38 


98341 


19831 


98010 


21559 


97648 


23260 


97257 


24954 


96837 


33 


18166 


9 8336 


19880 


98004 


21587 


97642 


23288 


9725i 


24982 


96829 


32 


n 


18195 


9 833i 


19908 


97998 


21616 


97636 


233i6 


97244 


25oio 


96822 


3i 


>J 


18224 


98325 


19937 


97992 


21644 


97630 


23345 


97237 


25o38 


9 68i5 


3c 


3i 


18252 


98320 


19965 


97987 


21672 


97623 


23373 


9723o 


25o66 


96807 


3 


3? 


1 8281 


983 1 5 


19994 


97981 


21701 


97617 


234oi 


97223 


23094 


96800 


33 


i83o 9 


983io 


20022 


97975 


21720 


97611 


23429 
23458 


97217 


25l22 


96793 


27 


A/j 


18338 


983o4 


2005l 


97969 
97 9 63 


21758 


97604 


97210 


25l5l 


96786 


26 


35 


l8 D 367 


98299 


20079 


21786 


97598 


23486 


97203 


25i7 9 


96778 


25 


36 


i83 9 5 


98294 


2010b 


97958 


21814 


97592 


235i4 


97196 


23207 


96771 


24 


37 


18424 


98288 


2oi36 


97932 


2i843 


97585 


23542 


97189 


25235 


96764 


23 


38 


18452 


98283 


20 1 65 


97946 


21871 


97579 


23571 


97i82 


25263 


96756 


22 


3 9 


18481 


98277 


20193 


97940 


21809 
21928 


97573 


23599 


97176 


20291 


96749 


21 


4o 


i85o 9 


98272 


20222 


979 3 4 


97566 


23627 


97169 


25320 


96742 


20 


4i 


i8538 


98267 


20250 


97928 


21956 


97560 


23656 


97162 


25348 


96734 


:i 


42 


18567 


98261 


20279 


97922 


21985 


97553 


23684 


97i55 


25376 


96727 


43 


i85 9 5 


98255 


20307 


97916 


220l3 


97547 


23712 


97148 


25404 


96719 


17 


44 


18624 


98250 


2o336 


97910 


2204I 


97341 


23740 


97141 


25432 


96712 


16 


45 


18652 


98245 


2o364 


97905 


22070 


97534 


23769 


97i34 


25460 


96705 


i5 


46 


1 868 1 


98240 


20393 


97890 
97893 


22098 


97528 


23797 


97127 


1 25488 


06697 


14 


% 


18710 


98234 


20421 


22126 


97521 


23825 


97120 


233l6 


96690 


i3 


i8 7 38 


98229 
98223 


2o45o 


97887 


22l55 


975 1 5 


23853 


97n3 


25543 


96682 


12 


4 9 


18767 


20478 


97881 


22183 


975o8 


23882 


97106 


25573 


96675 


II 


DO 


i8 79 5 


98218 


20007 


97875 


22212 


97502 


23910 


97100 


2D601 


96667 


IO 


5i 


18824 


98212 


2o535 


97869 
97863 


22240 


97496 


23938 


97093 


25629 


96660 


1 


52 


18852 


98207 


2o563 


22268 


97489 
97483 


23966 


97086 


25657 


96653 


53 


1 888 1 


98201 


20592 


97 85 7 


22297 


23995 


97079 


25685 


96645 


I 


54 


18910 


98196 


20620 


97851 


22325 


97476 


24023 


97072 


257i3 


96638 


55 


i8 9 38 


98190 


20649 


97845 


22353 


97470 


24o5i 


97065 


25741 


9663o 


5 


56 


18967 


9 8i85 


^0677 


97839 
97833 


22382 


97463 


24079 
24108 


97o58 


23769 
23798 


96623 


4 


ll 


18995 


9817- 


20706 


22410 


97457 


97o5i 


96615 


3 


19024 


98174 


20734 


97827 


22433 


9745o 


24i36 


97044 


25826 


96608 


1 


5g 


19052 


98168 


20763 


97821 


22467 


97444 


24164 


97o37 


25854 


96600 


1 


6o 


19081 


9 8i63 


20791 


97815 


2 2495 


97437 


24192 


97o3o 


25S82 


q65o3 





/ 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 




79° 


78° 


77° 


76° 


76° 


/ 



Table III. NATURAL SINES AND COSINES. 


67 


t 


15° 


16° 


17° 


18° 


19° 


/ 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 





25882 


96593 


27564 


96126 


29237 


9563o 


30902 


95lo6 


32557 


94552 


60 


I 


25910 


96585 


27592 


961 18 


29265 ! g5622 


30929 


95097 
9 5o88 


32584 


94542 


\% 


2 


25 9 38 


9 65 7 8 


27620 


961 10 


29293 


956i3 


30957 
3o 9 85 


32612 


94533 


3 


25966 


96570 


27648 


96102 


29321 


956o5 


95079 


3i63 9 


94523 


H 


4 


25994 


96562 


27676 


96094 


2 9 348 


95596 


3 1 1 2 


95070 


32667 


945 1 4 


5 


26022 


9 6555 


27704 


96086 


29376 


9 5588 


3 1 040 


95o6i 


32694 


945o4 


55 


6 


26o5o 


96547 


2773l 


96078 


29404 


95579 


3 1 068 


95o52 


32722 


944o5 


54 


7 


26079 


96540 


2 77 5 9 


96070 


29432 


95571 


31095 


95o43 


32749 


94485 


53 


8 


26107 


96532 


27787 


96062 


29460 


95562 


3ii23 


95o33 


32777 


94476 


52 


9 


2 6i35 


96524 


27813 


96o54 


29487 


95554 


3n5i 


95024 


32804 


94466 


5i 


10 


26i63 


96517 


27843 


96046 


295l5 


95545 


3h 7 8 


95oi5 


32832 


94457 


5o 


ii 


26191 


96509 


27871 


96037 


29543 


95536 


3 1 206 


95006 


3285 9 


94447 


% 


12 


26219 


96502 


27899 


96029 


2 9 5 7 l 


95528 


3i233 


94997 
94988 


32887 


94438 


i3 


26247 


96494 


27927 


96021 


29599 


95519 


3i26i 


32914 


94428 


47 


U 


26275 


96486 


27955 


96oi3 


29626 


955i 1 


31289 


94979 


32942 


944i8 


46 


i5 


263o3 


96479 


27983 


96oo5 


29654 


955o2 


3i3iO 


94970 


32969 


94409 


45 


16 


2633i 


96471 


2801 1 


95997 


29682 


95493 

9 5485 


3 1 344 


94961 


32997 


94399 


44 


17 


26359 


96463 


28039 


95989 


29710 


3i372 


94952 


33o24 


94390 
9438o 


43 


18 


2638 7 


96456 


28067 
28095 


95981 


1 2 9737 


95476 


3 1 399 


94943 


33o5i 


42 


'9 


26415 


96448 


95972 


29765 


95467 


3i42 7 


94933 


33079 


94370 


41 


20 


26443 


96440 


28123 


95964 


29793 


95459 


3i454 


94924 


33 1 06 


9436i 


40 


21 


26471 


96433 


28i5o 


95956 


29821 


95450 


31482 


94915 


33i34 


9435i 


\l 


22 


265oo 


96425 


28178 


95948 


29849 


95441 


3i5io 


94906 


33i6i 


94342 


23 


26528 


96417 


28206 


95940 


29876 


9 5433 


3i53 7 


94897 


33i8 9 


94332 


37 


24 


26556 


96410 


28234 


9593l 


29904 


95424 


3 1 565 


94888 


332i6 


9432 2 


36 


25 


26584 


96402 


28262 


95923 


29932 


954i5 


3i5g3 


94878 


33244 


943 1 3 


35 


26 


26612 


96394 


28290 


959l5 


29960 


95407 


31620 


94869 


33271 


943 o3 


34 


27 


.26640 


9 6386 


283 18 


95007 
95898 


29987 


9 53o8 


3i648 


94860 


33298 


942q3 


33 


28 


26668 


96379 


28346 


3ooi5 


95389 


31675 


9485 1 


33326 


94284 


32 


29 


26696 


96371 


28374 


95890 


3 oo43 


9 538o 


3 1 703 


94842 


33353 


94274 


3.i 


3o 


26724 


96363 


28402 


95882 


30071 


95372 


3i73o 


94832 


3338i 


94264 


3o 


3i 


26752 


9 6355 


28429 


95874 


30098 


9 5363 


3i 7 58 


94823 


334o8 


94254 


29 


32 


26780 


96347 


28457 


95865 


3oi26 


95354 


31786 


94814 


33436 


94245 


28 


33 


26808 


96340 


28485 


9 5857 


3oi54 


95345 


3i8i3 


948o5 


33463 


94235 


27 


34 


26836 


96332 


285i3 


95849 


3oi82 


95337 


3 1 841 


94795 


33490 


94225 


26 


35 


26864 


96324 


28541 


9^841 


30209 


95328 


3 1 868 


94786 


335i8 


942i5 


25 


36 


26892 


96316 


2856 9 


9 5832 


30237 


95319 


3 1896 


94777 


33545 


94206 


24 


37 


26920 


9 63o8 


285 97 


95824 


3o265 


953io 


31923 


94768 


33573 


94196 
94186 


23 


38 


26948 


963oi 


28625 


958i6 


30292 


953oi 


3i95i 


94758 


336oo 


22 


3 9 


26976 


96293 


28652 


95807 


3o32o 


952o3 


31979 


94749 


33627 


94176 


21 


4o 


27004 


96285 


28680 


95799 


3n348 


95284 


32006 


94740 


33655 


94167 


20 


4i 


27032 


96277 


28708 


95791 


30376 


95275 


32o34 


9473o 


33682 


94i57 


\l 


42 


27060 


96269 


28 7 36 


95782 


3o4o3 


95266 


32061 


94721 


33710 


94147 


43 


27088 


96261 


28764 


9 3 774 


3o43i 


95257 


32089 


94712 


33 7 3 7 


94i37 


17 


44 


27116 


96253 


28792 


95766 


30459 


95248 


32116 


94702 


33764 


94127 
941 18 


16 


45 


27144 


96246 


28820 


95757 


3o486 


95240 


32144 


94693 


33792 


i5 


46 


27172 


96238 


28847 


9 5 7 49 


3o5i4 


9523i 


32171 


94684 


338i 9 


94108 


14 


47 


27200 


96230 


28875 


9574o 


3o542 


95222 


32199 


94674 


33846 


94098 


i3 


43 


27228 


96222 


28903 


95732 


3o57o 


952i3 


32227 


94665 


33874 


94088 


12 


49 


27256 


96214 


28931 


95724 


30597 


95204 


32204 


94656 


33901 


94078 


11 


5o 


27284 


96206 


28959 


95715 


3o62D 


95195 
95i86 


32282 


94646 


33929 


94068 


10 


5i 


27312 


96198 


28987 


95707 


3o653 


32309 


94637 


33 9 56 


9 4o58 


8 


52 


27340 


96190 


29015 


95698 


3o68o 


95i77 


3 2 33 7 


94627 


33983 


94049 


53 


27368 


96182 


29042 


95690 
9 568i 


30708 


95168 


32364 


94618 


34oi 1 


94039 


7 


54 


27396 


96174 


29070 


3o736 


95159 


32392 


94609 


34o38 


94029 


6 


55 


27424 


96166 


29098 


956i3 


30763 


95i5o 


32419 


94599 


34o65 


94019 


5 


56 


27402 


96 1 58 


29126 


95064 


30791 


95142 


32447 


94590 


34093 


94009 


4 


57 


27480 


96i5o 


29154 


9 5656 


30819 


9Di33 


32474 


9458o 


34120 


93999 


3 


58 


27508 


96142 


29182 


95647 


3o846 


95124 


32502 


94571 


34147 


93989 


2 


5 9 


27536 


9 6i34 


29209 


95639 


30874 


931 15 


32529 


9456i 


34n5 


93979 


1 


60 


27564 


96126 


29237 


9563o 


3ogo2 


95io6 


32557 


94552 


34202 


93969 





_._. 


Cosine. 


Sine 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


' 




74° 


73° 


72° 


71° 


70° 



68 


NATURAL SINES AND COSINES. Table III 


f 


20° 


21° 


22° 


23° 


24° 


f 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 





34202 


93969 


3583 7 


93358 


37461 


92718 


39073 


92o5o 


40674 


91355 


60 


I 


34229 


93959 


35864 


93348 


37488 


92707 


39100 


92039 


40700 


91343 


11 


2 


34257 


93949 


358 9 i 


93337 


3 7 5i5 


92697 


39127 


92028 


40727 


9i33i 


3 


34284 


93939 


35918 


93327 


3 7 542 


92686 


39l53 


92016 


40753 


91319 


57 


4 


343 1 1 


93929 


35945 


933i6 


37569 
37595 


92675 


39180 


92005 


40780 


91307 


56 


5 


3433 9 


93919 


35973 


933o6 


92664 


39207 


91994 


40806 


91295 

91283 


55 


6 


34366 


93909 


36ooo 


93295 
93285 


37622 


92653 


39234 


91982 


40833 


54 


7 


343 9 3 


93899 


36027 


37649 


92642 


39260 


91971 


40860 


91272 


53 


8 


34421 


9 388 9 


36o54 


93274 


37676 


9263l 


39287 


9i 9 5 9 


40886 


91260 


52 


9 


34448 


93879 


36o8i 


93264 


37703 


92620 


393l4 


91948 


40913 


91248 


5i 


10 


34475 


93869 


36 1 08 


93253 


37730 


92609 


39341 


9ig36 


40939 


91236 


5o 


ii 


345o3 


9385 9 


36i35 


93243 


37757 


92598 


3 9 3d7 


91925 


40966 


91224 


% 


12 


3453o 


93849 


36i62 


93232 


37784 


92587 


39394 


91914 


40992 


91212 


i3 


34557 


9383g 


36190 


93222 


3781 1 


92576 


39421 


91902 


41019 


91200 


47 


14 


34584 


93829 


36217 


9321 1 


3 7 838 


92565 


3 9 448 


91891 


41045 


91188 


46 


i5 


34612 


93819 


36244 


93201 


3 7 865 


92554 


39474 


91879 


41072 


91176 


45 


16 


3463g 


93809 


36271 


93190 
93180 


37892 


92543 


39501 


91868 


41098 


91 164 


44 


17 


34666 


93799 
93789 


36298 


37919 


92532 


3 9 528 


9i856 


4U25 


9ii52 


43 


18 


34694 


36325 


93169 


07946 


92521 


39555 


91845 


4u5i 


91 140 


42 


J 9 


34721 


9^779 


36352 


93i 5o 


37973 


925 10 


39581 


9i833 


41178 


91 1 28 


4i 


20 


34748 


93769 


36379 


93148 


37999 


92499 
92488 


39608 


91822 


41204 


91116 


40 


21 


34775 


93739 


364o6 


93137 


38026 


39635 


91810 


4i23i 


91 104 


u 


22 


348o3 


93748 


36434 


9 3 1 27 


38o53 


92477 


3o66i 


91799 


41257 


91092 
91080 


23 


3483o 


93738 


3646i 


93i 16 


38o8o 


92466 


39688 


91787 


41284 


37 


24 


34857 


93728 


36488 


93106 


38107 


92455 


39715 


91773 


4i3io 


91068 


36 


25 


34884 


93718 


365 1 5 


93095 
93084 


38i34 


92444 


39741 


9n64 


4i337 


9io56 


35 


26 


34912 


93708 


36542 


. 38 1 6 1 


92432 


39768 


91752 


4 1 363 


91044 


34 


27 


34939 


93698 


3656 9 


93074 


38 1 88 


92421 


3 9 7 9 5 


91741 


4i3oo 


9io32 


33 


28 


34966 


93688 


365 9 6 


93o63 


382i 5 


92410 


39822 


91720 


41416 


91020 


32 


2Q 


34993 


93677 


36623 


93o52 


38241 


lllll 


39848 


91718 


41443 


91008 


3i 


3o 


35o2i 


93667 


3665o 


93c42 


38268 


39875 


91706 


41469 


90996 


3o 


3i 


35o48 


93657 


36677 


93o3i 


382 9 5 


92377 


39902 


91694 


41496 


90984 


29 


32 


35o75 


93647 


36704 


93020 


38322 


92366 


39928 


9 i683 


4l522 


90972 


28 


33 


35io2 


93637 


36 7 3i 


93oio 


38349 


92355 


39955 


91671 


4i54o 
41573 


90060 


27 


34 


35i3o 


93626 


36 7 58 


92999 
92988 


38376 


92343 


39982 


91660 


90948 


26 


35 


35i5 7 


93616 


36785 


384o3 


92332 


40008 


91648 


41602 


90936 


25 


36 


35 1 84 


93606 


368i2 


92978 


3843o 


92321 


4oo35 


9i636 


41628 


90924 


24 


ll 


352i 1 


93596 


3683 9 


92967 


38456 


923io 


40062 


91625 


4i655 


9091 1 


23 


35239 


93585 


36867 


92906 


38483 


92299 


40088 


9 i6i3 


41681 


90899 


22 


3o 


35266 


93575 


368 9 4 


92945 


385 10 


92287 


401 1 5 


91601 


41707 


90887 


21 


4o 


35393 


93565 


36921 


92935 


38537 


92276 


40141 


91590 


41734 


90875 


20 


4i 


35320 


93555 


36 9 48 


92924 


38564 


92265 


40168 


9.578 


41760 


90863 


» 


42 


35347 


93544 


36 97 5 


92913 


38591 


92254 


40195 


9 1 566 


41787 


9085 1 


43 


35375 


93534 


37002 


92902 


386i 7 


92243 


40221 


9i555 


4i8i3 


90839 


\l 


44 


35402 


93524 


37029 


92892 


38644 


9223 1 


40248 


9 i543 


41840 


90826 


45 


35429 


935i4 


37056 


92881 


38671 


92220 


40275 


9i53i 


41866 


90814 


i5 


46 


35456 


935o3 


37083 


92870 


38698 


92209 


40001 


91519 

91308 


41892 


90802 ! 14 


47 


35484 


93493 
93483 


37110 


92859 


38725 


92198 


4o328 


41919 


9°79<> 


i3 


48 


355u 


37i3 7 


92849 


38752 


92186 


4o355 


91496 
9U84 


41943 


90778 


12 


49 


35538 


93472 


37164 


92838 


38 77 8 
388o5 


92175 


4o38i 


41972 


90766 


11 


5o 


35565 


93462 


37191 


92827 


92164 


40408 


91472 


41998 


90753 


10 


5i 


35592 


93452 


37218 


92816 


38832 


92l52 


4o434 


91461 


42024 


90741 


I 


52 


35619 


93441 


37245 


92805 


3885 9 


92141 


40461 


9U49 


42o5i 


00729 


53 


35647 


9343i 


37272 


92794 
92784 


38886 


92i3o 


40488 


91437 


42077 


90717 


I 


54 


35674 


93420 


37299 


38 9 i2 


92119 


4o5i4 


9U25 


42104 


90704 


55 


35701 


93410 


37326 


92773 


38 9 3 9 


92107 


4o54i 


91414 


42i3o 


90692 


5 


56 


3D728 


93400 


37353 


92762 


38 9 66 


92096 
92085 


40567 


91402 


42 1 56 


90680 


4 


n 


3^55 


93389 


3738o 


92751 


38 99 3 


40594 


91390 


42i83 


00668 


3 


30782 


93379 


37407 


92740 


39020 


92073 


40621 


9 i3 7 8 


42209 


9o655 


2 


5 9 


358 10 


93368 


37434 


92720 
92718 


39046 


93062 


40647 


9 i366 


42235 


90643 


1 


6o 


35837 


93358 


37461 


39073 


9„o5o 


40674 


9i355 


42263 


9o63 1 





t 


Cosine. 


Sire. | 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


• 


69° 


68° 


67° 


66° j 


65° 



Tabus III. NATURAL SINES AND COSINES. 69 


/ 


__2[ 





26° 


27° 


28° 


29° 


/ 


Sine. 


Cosine. 


Sine. 


Cosine. 

89879 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


o 


42262 


9063 1 


43837 


45399 
45425 


89101 


46947 


88295 
88281 


48481 


87462 


60 


i 


42288 


90618 


43863 


89867 


89087 


46973 


485o6 


87448 


5o 


2 


423i5 


90606 


4388 9 


89854 


4545l 


89074 


46999 


88267 


48532 


87434 


58 


3 


42341 


90594 


43916 


89841 


45477 


89061 


47024 


88254 


4855 7 


87420 


5 7 


4 


42367 


90582 


43942 


89828 


455o3 


89048 


47o5o 


88240 


48583 


87406 


56 


5 


423 9 4 


90569 


43 9 68 


89816 


45529 


8 9 o35 


47076 


88226 


48608 


8 7 3 9 l 


55 


6 


42420 


90557 


43994 


89803 


45554 


89021 


47101 


88213 


48634 87377 


54 


I 


42446 


90545 


44020 


89790 


4558o 


89008 
88995 
88981 


47127 


88199 

88i85 


48659 


87363 


53 


42473 


90532 


44o46 


89777 


456o6 


47i53 


48684 


8734Q 
87335 


52 


9 


42499 
42525 


90520 


44072 


89764 


45632 


47178 


88172 


48710 


5i 


10 


90507 


44098 


89752 


45658 


88968 


47204 


88 1 58 


48 7 35 


87321 


5o 


ii 


42552 


90495 


44124 


80739 


45684 


88 9 55 


47229 
47255 


88144 


48761 


87306 


a 


12 


425 7 8 


90483 


44i 5i 


89726 


4571c 


88942 


88i3o 


48786 


87292 


i3 


42604 


90470 


44177 


89713 


45736 


88928 


47281 


88117 


4881 1 


87278 


47 


14 


4263 1 


90458 


442o3 


89700 


45762 


88 9 l5 


473o6 


88io3 


48837 


87264 


46 


i5 


42657 


90446 


44229 


89687 


45787 


88902 


47332 


88089 


48862 


87250 


45 


16 


42683 


90433 


44255 


89674 


458i3 


88888 


47358 


88075 


48888 


87235 


44 


\l 


42709 


90421 


44281 


89662 


45839 


888 7 5 


47383 


88062 


48913 


87221 


43 


42736 


90408 


443o7 


89649 


45863 


88862 


47409 


88048 


48 9 38 


87207 


42 


U) 


42762 


90396 


44333 


8 9 636 


458 9 i 


88848 


47434 


88o34 


48964 


87193 


4i 


20 


42788 


9 o383 


44359 


89623 


45917 


88835 


4746o 


88020 


48989 


87178 


4o 


21 


428i5 


90371 


44385 


89610 


45942 


88822 


47486 


88006 


49014 


87164 


39 


22 


42841 


9o358 


444i 1 


89597 


45968 


88808 


475i 1 


87993 


49040 


87l5o 


38 


23 


42867 


90346 


44437 


89584 


45994 


88795 


47537 


87979 
87965 


49065 


8 7 i36 


37 


24 


42894 


90334 


44464 


89571 


46020 


88782 


47562 


49090 


87121 


36 


25 


42920 


90321 


44490 


8 9 558 


46046 


88768 


47588 


87951 


491 16 


87107 


35 


26 


42946 


903 09 


445 16 


8 9 545 


46072 


88755 


47614 


87937 


49U1 


87093 


34 


27 


42972 


90296 


44542 


89532 


46097 


88741 


47639 


87923 


49166 


87079 
87064 


33 


28 


42999 
43o25 


90284 


44568 


89519 


46i23 


88728 


47660 


87909 
87896 


49192 


32 


29 


90271 


44594 


89506 


46149 
46175 


887 1 5 


47690 


49217 


87050 


3i 


3o 


43o5i 


90259 


44620 


89493 


88701 


47716 


87882 


49242 


87036 


3o 


3 1 


43077 


90246 


44646 


89480 


46201 


88688 


47741 


87868 


49268 


87021 


3 


32 


43 104 


90233 


44672 


89467 


46226 


88674 


47767 
47793 


87854 


49293 


87007 


33 


43 1 3o 


90221 


44698 


89454 


46252 


88661 


87840 


49318 


86 99 3 


27 


34 


43 1 56 


90208 


44724 


89441 


46278 


88647 


47818 


87826 


49344 


86978 


20 


35 


43i82 


90196 
90183 


447^0 


89428 


463o4 


88634 


47844 


87812 


49369 


86964 


25 


36 


43209 


44770 


89415 


4633o 


88620 


47869 


87798 


49394 


86949 


24 


12 


43235 


90171 
901 58 


44802 


89402 


46355 


88607 


47895 


87784 


49419 


86 9 3d 


23 


4326i 


44828 


8 9 38 9 


4638 1 


88593 
8858o 


47920 


87770 


49445 


86921 


22 


3 9 


43287 


90146 


44854 


89376 


46407 


47946 


87756 


49470 


86906 


21 


4o 


433i3 


90i33 


44880 


8g363 


46433 


88566 


47971 


87743 


49495 


86892 


20 


4i 


43340 


90120 


44906 


8 9 35o 


46458 


88553 


47997 


87729 
87715 


49521 


86878 


:i 


42 


43366 


90108 


44932 


8 9 33 7 


46484 


88539 


48022 


49546 


86863 


43 


43392 


90095 


44958 


8 9 324 


465 10 


88526 


48048 


87701 


49571 


86849 


17 


44 


43418 


90082 


44984 


8 9 3 1 1 


46536 


885i2 


48073 


87687 


49596 


86834 


16 


45 


43445 


90070 


45oio 


89298 


4656i 


88499 


48099 


87673 


49622 


86820 


i5 


46 


43471 


90057 


45o36 


89285 


46587 


88485 


1 48124 


8 7 65 g 


49647 


868o5 


14 


% 


43497 


90045 


45o62 


89272 


466i3 


88472 


48i5o 


87645 


4967 a 


86791 


1 3 


43523 


90032 


45o88 


8925.9 
89245 


46639 


88458 


48175 


8 7 63 1 


49697 


86777 


12 


49 


43549 


90019 


45u4 


46664 


88445 


48201 


87617 


49723 


86762 


u 


5o 


43575 


90007 
89994 


45 1 40 


89232 


46690 


8843i 


48226 


87603 


49748 


86748 


10 


5i 


436o2 


45 1 66 


89219 
89206 


46716 


88417 


48252 


8 7 58 9 
8 7 5 7 5 


49773 


86733 


I 


52 


43628 


89981 


45192 


46742 


88404 


48277 


49798 


86719 


53 


43654 


89968 


452i8 


89193 
89180 


46767 


883 9 o 


483o3 


87561 


49824 


86704 


1 


54 


4368o 


89956 


45243 


46793 


883 77 


48328 


87546 


49849 


86690 


6 


55 


43706 


89943 


4526q 
45293 


89167 


46819 


88363 


48354 


8 7 532 


49874 


86675 


5 


56 


43733 


89930 


89153 


46844 


88349 


483 7 o 
484o5 


87618 


49899 


86661 


4 


n 


43759 


89918 


45321 


89140 


46870 


88o36 


87504 


49924 


86646 


3 


43 7 8a 
438n 


89905 


45347 


89127 


46896 


88322 


4843o 


87490 


49950 


86632 


2 


5 9 


89892 


453 7 3 


89114 


46921 


883o8 


48456 


87476 


49975 


86617 


1 


6o 


43837 


89879 


45399 


89101 


46947 


88295 


48481 


87462 


5oooo 


866o3 







Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


t 


/ 


64° 


63° 


62° 


61° 


60° 



70 NATURAL SINES AND COSINES. Table III. 


/ 




30° 


81° 


32° 


33° 


34° 




Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 




5oooo 


866o3 


5i5o4 


85 7 I 7 


52992 


848o5 


54464 


83867 


55919 


82904 60 
82887 5q 


i 


5oo25 


86588 


5i5^9 


85702 


53oi7 


84789 


54488 


8385i 


55 9 43 


2 


5oo5o 


865 7 3 


5 1 554 


8568 7 


53o4l 


84774 


545 1 3 


83835 


55968 


82871 


58 


3 


50076 


86559 


51679 


856 7 2 


53o66 


84759 


54537 


838io 


55992 


82855 


57 


4 


5oioi 


86544 


5 1 604 


85657 


53091 


84743 


5456 1 


838o4 


56oi6 


82839 


56 


5 


5oi26 


8653o 


5i628 


85642 


53u5 


84728 


54586 


83788 


56o4o 82822 


55 


6 


5oi5i 


865i5 


5i653 


856^7 


53 1 40 


84712 


54610 


83 77 2 


56o64 j 82806 


54 


I 


50176 


865oi 


5i6 7 8 


85bi2 


53 1 64 


84697 


54635 


83 7 56 


56o88 82790 


53 


50201 


86486 


5i7o3 


855 9 7 


53189 


84681 


54659 
54683 


83740 


56na 1 82773 


52 


9 


50227 


86471 


51728 


85582 


53214 


84666 


83 7 24 


56i36 1 82757 


5i 


;c 


50202 


86457 


5i 7 53 


85567 


53238 


8465o 


54708 


83 7 o8 


56i6o , 82741 


5o 


ii 


50277 


86442 


5i 77 8 


8555i 


53263 


84635 


54732 


836 g2 


06184 82724 


% 


12 


5o3o2 


86427 


5i8o3 


85536 


53288 


84619 


54756 836 7 6 


56208 i 82708 


i3 


5o327 


864 1 3 


51828 


85521 


533i2 


84604 


54781 


8366o 


56232 


82692 


47 


U 


5o352 


863 9 8 


5i852 


855o6 


53337 


84588 


548o5 


83645 


56256 


82675 


46 


i5 


5o377 


86384 


5i8 77 


85491 


5336i 


84573 


54829 


8362 9 


5628o 


82659 


45 


16 


5o4o3 


8636 9 


51902 


85476 


53386 


84557 


54854 


836i3 


563o5 


82643 44 


\l 


5o428 


86354 


01927 


8546i 


534i 1 


84342 


54878 


835 97 
8358i 


56329 


82626 


43 


5o453 


86340 


51952 


85446 


53435 


84526 


54902 


56353 


82610 


42 


l 9 


50478 


86325 


51977 


8543 1 


5346o 


845 1 1 


54927 


83565 


56377 


825o3 


4i 


20 


5o5o3 


863 10 


52002 


854i6 


53484 


844o5 


54951 


83549 


564oi 


8 2 5 77 


4o 


21 


5o528 


86295 


52026 


854oi 


535o9 


84480 


54975 


83533 


56425 


8256i 


3 9 


22 


5o553 


86281 


52o5i 


85385 


53534 


84464 


54999 


835i 7 


56449 


82544 


38 


23 


50578 


86266 


52076 


85370 


53558 


84448 


55o24 


835oi 


56473 


82528 


37 


24 


5o6o3 


862 5 1 


52101 


85355 


53583 


84433 


55o48 


83485 


56497 


82DII 


3b 


25 


50628 


86237 


52126 


8534o 


536o7 


84417 


55072 


8346 9 


56521 


82495 


35 


26 


5o654 


86222 


52i5i 


85325 


53632 


84402 


55097 


83 4 53 


56545 


82478 


34 


27 


50679 


86207 


52175 


853 10 


53656 


84386 


55i2i 


83 4 3 7 


56569 


82462 


33 


28 


50704 


86192 


52200 


85294 


5368 1 


84370 


55i45 


83421 


565g3 


82446 


32 


29 


50729 


86178 


52225 


83279 


53705 


84355 


55169 


834o5 


56617 


82429 
82413 


3i 


3o 


50754 


86 1 63 


52250 


85264 


5373o 


84339 


55194 


8338 9 


56641 


3o 


3i 


50779 


86148 


52275 


85249 


53754 


84324 


552i8 


833 7 3 


56665 


82396 

8238o 


3 


32 


5o8o4 


86i33 


52299 


85234 


53779 


843o8 


55242 


83356 


5668 9 


33 


50829 


861 19 


52324 


852i8 


538o4 


84292 


53266 


83340 


56713 


82363 


27 


34 


5o854 


86104 


52349 


852o3 


53828 


84277 


55291 


83324 


56736 


82347 


26 


35 


50879 


86089 


5 2 3 7 4 


85 188 


53853 


84261 


553i5 


833o8 


56i6o 


8233o 


25 


36 


50904 


86074 


52399 
52423 


85 173 


538 77 


84245 


55339 


83292 


56 7 84 


823i4 


24 


ll 


50929 


86059 


85i57 


53902 


8423o 


55363 


832 7 6 


568o8 


82297 


23 


50954 


86o45 


52448 


85 1 42 


53926 


84214 


55388 


8326o 


56832 


82281 


22 


3 9 


50979 


86o3o 


52473 


83127 


53961 


84198 


55412 


83244 


56856 


82264 


21 


40 


5 1 004 


86oi5 


52498 


85ii2 


53975 


84182 


55436 


83228 


5688o 


82248 


20 


4i 


51029 


86000 


52022 


85096 


54ooo 


84167 


55460 


83212 


56904 


8223l 


19 


42 


5io54 


85 9 85 


52347 


85o8i 


34024 


84131 


33484 


83i 9 5 


56928 


82214 


l8 


43 


51079 


85970 


52372 


85o66 


34049 


84i35 


555o9 


83i 79 


56932 


82198 


17 


44 


5uo4 


80936 


52597 


85o5i 


54073 


84120 


55533 


83 1 63 


56976 


82181 


16 


45 


51129 


85941 


52621 


85o35 


54097 


84104 


55557 


83 1 47 


57000 


82160 


13 


46 


5n54 


85926 


52646 


83020 


54122 


84088 


5558i 


S3i3i 


5 7024 


82148 


14 


% 


51179 


85qii 


52671 


85oo5 


54146 


84072 


556o5 


83n5 


57047 


82i32 


i3 


5i2o4 


85896 


52696 


84989 


54171 


84057 


5563o 


83o 9 8 
83o82 


07071 


82ii5 


12 


49 


51229 


8588i 


52720 


84974 


54195 


84041 


55654 


57095 


82098 
82082 


11 


5o 


5i254 


85866 


52745 


84959 
84943 


54220 


84025 


55678 


83o66 


57119 


IO 


5i 


51279 


8585i 


52770 


54244 


84009 


55702 


83o5o 


5 7 i43 


82065 


i 


52 


5i3o4 


85836 


52794 


84928 


54269 
542 9 3 


83994 


55726 


83o34 


57167 


82048 


53 


5i32g 


85821 


52819 


84913 
84897 


83 97 8 


5575o 


83oi7 


57191 


82032 


7 


54 


5i354 


858o6 


52844 


543i7 


83 9 62 


55 77 5 


83 00 1 


5-210 


82015 


6 


55 


5i379 


85 79 2 


52869 


84882 


54342 


83 9 46 


55 799 


82985 


57238 


81999 
81982 


5 


^6 


5 1 404 


85 777 


52893 


84866 


54366 


8393o 


55823 


82969 
82953 


57262 


4 


n 


51429 


85762 


52918 


8485i 


543 9 i 


83 9 i5 


55847 


5-286 


81960 


3 


5 1 454 


85 7 47 


52943 


84836 


544i5 


838 99 
83883 


558 7 i 


82 9 36 


57310 


Si 949 


2 


5 9 


5 1 479 


85732 


52967 


84820 


54440 


55890 


82920 


5;334 


81932 


1 


60 


5i5o4 


85 7 i 7 


52992 


84803 


54464 


83867 


55919 


82904 


57358 


8i 9 i5 





t 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


/ 


5»' J 


68 3 


67° 


66° 


55° 



Table IIL NATURAL SINES AND COSINES. 71 


' 


35° 


36° 


37° 


38° 


39° 


60 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


o 


57358 


81915' 
81899 . 

81882 


58 779 


80902 


60182 


79864 


6 1 566 


78801 


62932 


77 7 l5 


i 


5 7 38i 


58802 


8o885 


6o2o5 


79846 


6i58 9 


7 8 7 83 


62955 


77696 


% 


2 


57405 


58826 


80867 


60228 


79829 


61612 


7 8 7 65 


62977 


77678 


3 


57429 
5745J 


8 1 865 


58849 


8o85o 


6o25i 


79811 


6i635 


78747 


63ooo 


77660 


5 7 


4 


81848 


58873 


8o833 


60274 


7979 3 


6i658 


78729 


63022 


77641 


56 


5 


57477 


8i832 


588 9 6 


80816 


60298 


79776 


61681 


787,1 


63o45 


77623 


55 


6 


57501 


8i8i5 


58920 


80799 
80782 


6o32i 


797 58 


61704 


78694 


63o68 


776o5 


54 


I 


57524 


81798 
81782 


58943 


6o344 


79741 


61726 


78676 


63090 


7 7 586 


53 


57548 


58967 


80765 


60367 


79723 


61749 


78658 


63u3 


77568 


52 


9 


5 7 5 72 


8n65 


5899a 


80748 


60390 


79706 


61772 


78640 


63i35 


7755o 


5i 


10 


57596 


81748 


59014 


80730 


60414 


79688 


61795 


78622 


63 1 58 


7 7 53 1 


5o 


ii 


57619 


8i 7 3i 


5go37 


8o 7 i3 


60437 


79671 


61818 


78604 


63 1 80 


775i3 


3 


12 


57643 


81714 


59061 


80696 


60460 


79653 


61841 


78586 


632o3 


77494 


i3 


57667 


81698 
81681 


59084 


80679 


6o483 


79635 


61864 


7 8568 


63225 


77476 


47 


14 


57691 


59108 


80662 


6o5o6 


79618 


61887 


*)855o 


63248 


77458 


46 


i5 


57715 


81664 


59131 


80644 


60529 


79600 


61909 


78532 


63271 


77439 


45 


16 


5 77 38 


81647 


59154 


80627 


6o553 


7 9 583 


61932 


78514 


63293 


77421 


44 


17 


57762 


8 1 63 1 


59178 


80610 


60576 


79565 


61955 


78496 


633 1 6 


77402 


43 


18 


57786 


81614 


59201 


80393 


60599 


79547 


61978 


78478 


63338 


77384 


42 


'9 


57810 


8i5 97 


59225 


80576 


60622 


7953o 


62001 


78460 


6336i 


77366 


41 


20 


5 7 833 


8i58o 


59248 


8o558 


6o645 


7 9 5 1 2 


62024 


78442 


63383 


77347 


40 


21 


5 7 85 7 


8 1 563 


59272 


8o54i 


60668 


79494 


62046 


78424 


634o6 


77329 


3 9 


22 


57881 


81 546 


59295 


8o524 


60691 


79477 


62069 


78405 


63428 


773io 


38 


23 


57904 


8i53o 


5 9 3i8 


80507 


60714 


79459 


62092 


78387 


6345i 


77292 


37 


24 


57928 


8i5i3 


59342 


80489 


60738 


794^1 


621 15 


78369 


63473 


77273 


36 


25 


57952 


81496 


59365 


80472 


60761 


79424 


6 2 i38 


7 835i 


63496 


77255 


35 


26 


57976 


81479 


5 9 389 


8o455 


60784 


79406 


62160 


78333 


635i8 


77236 


34 


2 


57999 


81462 


5 9 4i2 


8o438 


60807 


79388 


6ai83 


783 1 5 


6354o 


77218 


33 


58023 


8i445 


5 9 436 


80420 


6o83o 


79371 


62206 


78297 


63563 


77199 


32 


29 


58047 


81428 


59459 


8o4o3 


6o853 


79353 


62229 


78279 


63585 


77181 


3i 


3o 


58070 


81412 


59482 


8o386 


60876 


79335 


6225l 


78261 


636o8 


77162 


3o 


3i 


58094 


8i3 9 5 


59506 


8o368 


60899 


79318 


62274 


78243 


6363o 


77144 


29 


32 


58n8 


81378 


59529 


8o35i 


60922 


79300 


62297 


78225 


63653 


77120 


28 


33 


58i4i 


8i36i 


59552 


8o334 


60945 


79282 


62320 


78206 


63675 


77107 


27 


34 


58 1 65 


8i344 


59576 


8o3i6 


60968 


79264 


62342 


78188 


636 9 8 


77088 


26 


35 


58189 


8i3a 7 


59599 


80299 


60991 


79^47 


62365 


78170 


63720 


77070 


25 


36 


58212 


8i3io 


59622 


80282 


6ioi5 


79229 


62388 


78152 


63742 


77o5i 


24 


37 


58236 


81293 


59646 


80264 


6io38 


792 1 1 


62411 


78134 


63765 


77o33 


23 


38 


58260 


81276 


59669 
59693 


80247 


61061 


79193 


62433 


78116 


63 7 8 7 


77 OI 4 


22 


3 9 


58283 


812D9 


8o23o 


61084 


79176 


62456 


78098 


638io 


76996 


21 


40 


58307 


81242 


59716 


80212 


61107 


7 9 i58 


62479 


78079 


63832 


76977 


20 


4i 


5833o 


81225 


5g73o 
59763 


8oi 9 5 


6u3o 


79140 


62502 


78061 


63854 


76909 


IS 


42 


58354 


81208 


80178 


6n53 


79122 


62524 


78043 


638 77 


76940 


43 


58378 


81191 


59786 


80160 


61176 


79105 


62547 


78025 


638 99 


76921 


17 


44 


584oi 


81174 


59809 


8oi43 


61199 


79087 


62570 


78007 


63922 


76903 


16 


45 


58425 


8u5 7 


5 9 832 


8oi25 


61222 


79069 


62592 


77988 


63944 


76884 


i5 


46 


58449 


81 1 40 


5 9 856 


80108 


6i245 


7905 1 


62615 


77970 


63 9 66 


76866 


14 


47 


58472 


8ii23 


59879 


80091 


61268 


79o33 


62638 


77932 


63989 


76847 


i3 


48 


58496 


81 106 


59902 


80073 


61 291 


79016 

78998 
78980 


62660 


779 3 4 


640 1 1 


76828 


12 


49 


585i 9 


81089 


59926 


8oo56 


6i3i4 


62683 


77916 


64o33 


76810 


11 


5o 


58543 


81072 


59949 


8oo38 


6i337 


62706 


77897 


64o56 


76791 


10 


5i 


5856 7 


8io55 


59972 


80021 


6i36o 


78962 


62728 


77879 


64078 


76772 


% 


52 


585 9 o 


8io38 


59995 


8ooo3 


6i383 


78944 


62751 


77861 


64100 


76754 


53 


586 1 4 


81021 


60019 


79986 


61406 


78926 


62774 


77843 


64123 


76735 


7 


54 


58637 


81004 


60042 


79968 


61429 


78908 


62796 


77824 


64145 


76717 


6 


55 


5866i 


80987 


6oo65 


799 5i 


6i45i 


78891 


62819 


77806 


64167 


76698 


5 


56 


58684 


80970 


60089 


79934 


6i474 


78873 


62842 


77788 


64190 


76679 


4 


57 


58708 


80953 


60112 


79916 
79899 


61497 


78855 


62864 


77769 


64212 


76661 


3 


58 


58 7 3i 


809 36 


6oi35 


6i52o 


7 883 7 


62887 


77751 


64234 


76642 


2 


5 9 


58755 


80919 


60 1 58 


79881 


61 543 


78819 


62909 


77733 


64256 


76623 


1 


60 


58779 


80902 


60182 


79864 


6 1 566 


78801 


62932 


77715 


64279 


76604 





t 


Cosine 


Sine. 


Cosina. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


/ 


54° 


53° 


52° 


51° 


50° 



72 NATURAL SINES AND COSINES. Table III. 


/ 


40° 


41° 


42° 


43° 


44° 


/ 


Sine. 


Cosine. 


Sine. ! Cosine. 


Sine. 


Cosine. 


Sine. 1 Cosine. 


Sine. 


Cosine. 





64279 


76604 


656o6 


70471 


66913 


743 1 4 


68200 


73i35 


69466 


71934 


60 


i 


643oi 


76586 


65628 


75452 


66930 


742 9 5 


j 68221 


73u6 


69487 


71914 


is 


2 


64323 


76567 


6565o 


75433 


66906 


74276 


68242 


73096 


69508 


71894 


3 


64346 


76548 


65672 


704U 


66978 


74206 


! 68264 


73076 


69529 


71873 


57 


4 


64368 


7653o 


65694 


753 9 5 


66999 


7423 7 


1 68285 


73oo6 


69549 


71803 


56 


5 


64390 


765i 1 


65716 


7 53 7 


67021 


74217 


' 683o6 


73o36 


69570 


71833 


55 


6 


64412 


76492 


65738 


75356 


67043 


74198 


1 68327 


73oi6 


69591 


7i8i3 


54 


7 


64435 


76473 


65759 


75337 


67064 


74178 


68349 


72996 


69612 


71792 


53 


8 


64457 


76455 


65781 


7 53i8 


67086 


74i59 


; 68370 


72976 


6 9 633 


7H72 


52 


9 


64479 


76436 


658o3 


75299 
75280 


67107 


74i3g 


I 683 9 i 


72907 


69654 


71702 


5i 


10 


645oi 


76417 


65825 


67129 


74120 


68412 


72937 


69675 


71732 


So 


ii 


64024 


76398 


65847 


75261 


67151 


74100 


68434 


72917 


69696 


71711 


3 


12 


64546 


7638o 


6586g 


7524i 


67172 


74080 


68455 


72897 


69717 


71691 


i3 


64568 


7636i 


65891 


75222 


67194 


74061 


68476 


72877 


69737 


71671 


47 


14 


64090 


76342 


60913 


752o3 


67215 


74o4i 


68497 


7 285 7 


69758 


7i65o 


46 


i5 


64612 


76323 


65935 


75184 


67237 


74022 


685i8 


7 283 7 


69779 


7i63o 


45 


16 


64635 


763o4 


65956 


75i65 


6 72 58 


74002 


6853 9 


72817 


69800 


71610 


44 


17 


64657 


76286 


65978 


75i46 


67280 


73983 


68061 


72797 


69821 


7i5go 


43 


18 


64679 


76267 


66000 j 75126 


67301 


73963 


68082 


72777 


69842 


7i56 9 


A* 


J 9 


64701 


76248 


66022 


70107 


67323 


73q44 


686o3 


72757 


69862 


7i549 


4i 


20 


64723 


76229 


66044 


75o88 


67344 


73924 


68624 


72737 


6 9 883 


7l52Q 


4o 


21 


64746 


76210 


66066 


75069 


67366 


73904 
73885 


68645 


72717 


69904 


7i5o8 


39 


22 


64768 


76192 


66088 


75o5o 


67387 


68666 


72697 


69925 


71488 


38 


23 


64790 


76173 


66109 


70o3o 


67409 


73865 


68688 


72677 


69946 


71468 


37 


24 


64812 


76i54 


66i3i 


75ou 


67430 


73846 


68709 


72657 


69966 


71447 


36 


23 


64834 


76i35 


66 1 53 


74992 


67452 


73826 


68730 


72637 


69987 


71427 


35 


26 


64856 


76116 


66175 


74973 


67473 


73806 


68 7 5i 


72617 


70008 71407 


34 


2 7 


64878 


76097 


66197 


74953 


67495 


73787 


68772 


72597 


70029 


71386 


33 


23 


64901 


76078 


66218 


74934 


67516 


73767 


68793 


72077 


70049 


7 1 366 


32 


29 


64923 


76059 


66240 


749i5 


67538 


73747 
73728 


68814 


72007 


70070 


71340 j 3i 


3o 


64945 


76041 


66262 


74896 


67559 


68835 


72537 


70091 


7l32D 


3o 


3i 


64967 


76022 


66284 


74876 


67580 


73708 


68857 


72517 


70II2 


7i3o5 


3 


32 


64989 


t6oo3 


663o6 


74857 


67602 


73688 


68878 


72497 


7oi32 


71284 


33 


65ou 


759&l 


66321 


74838 


67623 


73669 


68899 


72477 


70 1 53 


71264 


27 


34 


65o33 


7 5 9 65 


66349 


74818 


67645 


73649 


68920 


72407 


70174 


71243 


26 


35 


65o55 


73946 


66371 


74799 


67666 


73629 


! ^941 


72437 


70190 


71223 


25 


36 


65o77 


75927 


663 9 3 


74780 


67688 


736io 


68962 


724:7 


702 1 5 


7i2o3 


24 


37 


65ioc 


73908 


664i4 


74760 


67709 


73590 


68983 


723Q7 


70236 


71182 


33 


38 


65l22 


75889 


66436 


74741 


67730 


73570 


69004 


72377 


70257 


71162 


22 


3 9 


65 1 44 


75870 


66458 


74722 


67702 


7355i 


69025 


72307 


70277 


7ii4i 


21 


4o 


65 1 66 


7535i 


66480 


747o3 


67773 


7353i 


69046 


72337 


70298 


71121 


30 


4i 


65i88 


75832 


665oi 


74683 


67793 


735u 


69067 


7 23i 7 


70319 


71100 


)t 


42 


652io 


758i3 


66523 


74664 


67816 


73491 


69088 


72297 


70339 


71080 


43 


65232 


75794 


66545 


74644 


67837 


73472 


69109 


72277 


7o36o 


71059 


17 


44 


65254 


75 77 5 


66566 


74625 


67859 


73452 


69130 


72207 


7o38i 


71039 


16 


45 


65276 


7^756 


66588 


74606 


67880 


73432 


69101 


72236 


70401 


71019 


i5 


46 


65298 


75 7 38 


66610 


74586 


67901 


734i3 


69172 


72216 


70422 


10098 


14 


47 


65320 


75719 


66632 


7456 7 


67923 


73393 


69193 


72196 


70443 


70978 


i3 


48 


65342 | 75700 


66653 


74548 


67944 


7 33 7 3 


69214 


72176 


70463 


70907 


12 


49 


65364 | 75o8o 


66675 


74528 


67965 


73353 


69235 


72i56 


70484 


70937 


11 


5o 


65386| 75661 


66697 


74509 


67987 


73333 


69206 


72i36 


7o5o5 


70916 
70896 


10 


5i 


654o8 75642 


66718 


74489 


68008 


733i4 


69277 


72116 


70520 


I 


52 


6543o ! 75623 


66740 


74470 


68029 


73294 


69298 


72095 


70546 


7 o8 75 


53 


65452 


70604 


66762 


7445 1 


68o5i 


73274 


69319 


7 207 5 


70567 


70800 


7 


54 


65474 


75585 


66783 


7443 1 


68072 


73254 


69340 


72o55 


70587 


70834 


6 


55 


654g6 


75566 


66800 


74412 


68o 9 3 


73234 


6 9 36i 


72o35 


70608 


70813 


5 


56 


655i8 


75547 


66827 


74392 


681 15 


732i5 


6 9 382 


72015 


70628 


7°79 3 


4 


57 


6554o 


75528 


66848 


743 7 3 


68i36 


73195 


69403 


7 1 990 


70649 


70772 


3 


58 


65562 


75509 


66870 


74353 


6S07 


73i75 


69424 


71974 


70670 


70702 


2 


5 9 


65584 


75490 


66891 


74334 


68179 


73i 55 


69445 


71904 


70690 


70731 


1 


6o 


656o6 


75471 


66913 


743i4 


68200 


73i35 


69466 


71934 


70711 


70711 





t 


Cosine. 1 Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sine. 


Cosine. 


Sina. 




49° 


48° 


47° 


46° 


45° 





Table III. NATURAL TANGENTS AND COTANGENTS. 73 


i 


0° 


1° 


2° 


3° 


t 
60 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangeat. Cotang. 


o 


00000 


Infinite. 


OI746 


57 • 2900 


03492 


28-6363 


o524l iq.0811 


i 


00029 
ooo58 


3437^5 
1718.87 


01775 


56 


35o6 


03521 


28-3994 


05270 




•9755 

871 1 


\l 


2 


Ol8o4 


55 


44i5 


o355o 


28-1664 


05299 


18 


3 


00087 


II45.92 


oi833 


54 


56i3 


03579 


27-9372 


o532§ 


18 


7678 


n 


4 


001 16 


85 9 


436 


01862 


53 


708G 


03609 


27.7117 


o5357 


18 


6656 


5 


00145 


687 


549 


01891 


52 


8821 


o3638 


27.4899 


05387 


18 


5645 


55 


6 


00173 


572 


957 


01920 


52 


0807 


03667 


27.2715 


o54i6 


18 


4645 


54 


7 


C0204 


491 


106 


01949 
01978 


5i 


3o32 


03696 


27-o566 


o5445 


18 


3655 


53 


8 


00233 


429 


718 


5o 


5485 


c3725 


26-845o 


o5474 


18 


2677 


52 


9 


00262 


38i 


97' 


02007 


4q 


• 8i5 7 


03754 


26-6367 


o55o3 


18 


1708 


5i 


IO 


00291 


343 


521 


02o36 


49 


1039 


o3 7 83 


26-43i6 


o5533 


18 


0750 


5o 


i i 


00320 


3l2 


02066 


48 


• 4121 


o38i2 


26-2296 


o556 2 


17 


9802 
8863 


it 


12 


00349 


286 


478 


02095 


47 


7395 


o3842 


26 c'507 


05391 


17 


i3 


00378 


264 


441 


02124 


47 


o853 


03871 


25 8348 


o5620 


H 


79 34 


47 


14 


00407 


245 


552 


02 1 53 


46 


4489 


03900 


25-64i8 


o5649 

05678 


17 


7oi5 


46 


i5 


oo436 


229 


182 


02182 


45 


8294 


03929 


25-45i7 


17 


6106 


45 


16 


00465 


214 


858 


02211 


45 


2261 


o3 9 58 


25-2644 


05708 


'7 


52o5 


44 


17 


00495 


202 


219 


02240 


44 


6386 


03987 


25-0798 


05737 


17 


43i4 


43 


18 


oo524 


190 


984 


02269 
02298 


44 


0661 


04016 


24-8978 


05766 


»7 


3432 


42 


19 


oo553 


180 


9 32 


43 


5o8i 


04046 


24-7i85 


05795 


17 


2558 


4i 


2C 


oo582 


171 


885 


02328 


42 


9641 


04075 


24.5418 


o5824 


17 


1693 

o83 7 


40 


21 


0061 1 


1 63 


700 


02357 


42 


4335 


04104 


24.3675 


o5854 


H 


18 


22 


00640 


1 56 


259 


02386 


4! 


9 i58 


04 1 33 


24-1957 


o5883 


16 


9990 


23 


00669 
00698 


149 

143 


463 


024 1 5 


4i 


4106 


04162 


24.0263 


05912 


16 


9130 


37 


24 


23 7 


02444 


40 


9H4 


04191 


23-85 9 3 


05941 


16 


83i9 


36 


25 


00727 


i3 7 


507 


02473 


40 


4358 


04220 


23-6945 


05970 


16 


7496 


35 


26 


00756 


132 


219 


025o2 


3q 


o655 
3059 
o568 


04230 


23-5321 


05999 


16 


6681 


34 


27 


00785 


127 


321 


0253i 


3 9 


04279 

o43o8 


23.3718 


06029 


16 


58 7 4 


33 


28 


00814 


122 


774 


o256o 


3 9 


23-2i37 


060 5 8 


16 


5o 7 5 


32 


29 


00844 


118 


540 


02589 


38 


6177 


04337 


23-o577 


06087 


16 


4283 


3i 


3o 


00873 


114 


58 9 


02619 


38 


1885 


04366 


22>9o38 


061 16 


16 


3499 


3o 


3i 


0O902 


no 


892 


02648 


37 


7686 


04395 


22-7519 


o6i45 


16 


2722 


29 


32 


00931 


107 


426 


02677 


3 7 


35 79 


04424 


22-6020 


06175 


16 


1952 


28 


33 


00960 


104 


Hi 


02706 


36 


g56o 
5627 


04454 


22-4541 


06204 


16 


1 190 


27 


34 


00989 


101 


107 


02735 


36 


04483 


22-3o8l 


o6233 


16 


0435 


26 


35 


01018 


98-2179 


02764 


36 


1776 


045 1 2 


22-1640 


06262 


i5 


9687 
8945 


25 


36 


01047 


95-4893 


02793 


35 


8006 


04541 


22-0217 


06291 


i5 


24 


u 


01076 


92-9085 


02822 


35 


43i3 


04570 


2i-88i3 


o632i 


i5 


8211 


23 


ono5 


90-4633 


0285i 


35 


069 5 
71 5i 


04599 


21-7426 


o635o 


i5 


7483 


22 


3 9 


on35 


88-1436, 


02881 


34 


04628 


2i-6o56 


06379 


i5 


6762 


21 


4o 


01164 


85- 9 3 9 8 
83-8435 


02910 


34 


36 7 8 


o4658 


21-4704 


06408 


i5 


6048 


20 


4i 


01 193 


02939 
02968 


34 


0273 


04687 


21-3369 


06437 


]5 


534o 


;g 


42 


01222 


81-8470 


33 


6935 


04716 


2 1 - 2049 


06467 


i5 


4638 


43 


OI25l 


79-9434 
78- 1263 


02997 


33 


3662 


04745 


21-0747 


06496 


i5 


3943 


\l 


44 


01280 


o3o26 


33 


0452 


04774 


20-9460 


o6525 


i5 


3254 


45 


01309 


76-3900 


o3o55 


32 


73o3 


o48o3 


20-8188 


o6554 


i5 


2571 


i5 


46 


oi338 


74-7292 


o3o84 


32 


42i3 


04832 


20-6q32 


o6584 


(5 


i8 9 3 


14 


47 


01367 


73- 1390 


o3u4 


32 


1181 


04862 


20.5691 


066 1 3 


i5 


1222 


i3 


48 


01396 


7i-6i5i 


o3i43 


3. 


82o5 


04891 


20-4465 


06642 


i5 


o55 7 


12 


4 9 


01425 


7o«i533 


03172 


3. 


5284 


04920 


20 3253 


06671 


14 


9898 


1 1 


5o 


oi455 


68.7501 


03201 


3. 


2416 


04949 


2o-2o56 


06700 


14 


9244 


10 


5i 


01484 


67.4019 


o323o 


3o 


9 5 99 


04978 


20-0872 


06730 


14 


8596 


I 


52 


oi5i3 


66-io55 


o3259 
o3288 


3o 


6833 


03007 


19-9702 
19-8546 


06759 
06788 


H 


7934 


53 


01 542 


64-858o 


3o 


4116 


o5o3t 


14 


7317 


7 


54 


01571 


63-6567 


o33i7 
o3346 


3o 


1446 


o5o66 


19-7403 


06817 


14 


6685 


6 


55 


01600 


62-4992 


2 9 


8823 


o5o95 


19-6273 


06847 


14 


6059 


5 


56 


01629 


61-3829 
6o-3o58 


03376 


29 


6245 


o5i24 


I9«5i56 


06876 


14 


5438 


4 


t 


oi658 


o34o5 


29 


3 7 n 


o5i53 


I9-4o5i 


06903 


1 4 


4823 


3 


5P j 01687 


5o-2659 
58-26i2 


o3434 


29 


1220 


o5i82 


19-2959 
19.1879 


06934 


14 


4212 


2 


5^ j 01716 


o3463 


28- 


8771 


05212 


06963 


14 


3607 


1 


60 i 01746 


57.2900 


03492 


28-6363 


o524i 


19.0811 


06993 


14 


3007 




t 


/ 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Ta 


igent. 


89° 


88° 


87° 


86° 



74 NATURAL TANGENTS AND COTANGENTS. Table III. 


/ 


4 


-° 


5° 


6° 


7° 


/ 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 





06993 


l4-3oo7 


08749 

08778 


n-43oi 


Io5io 


9'5i436 


12278 


8-14435 


60 


i 


07022 


14 


■2411 


11 -3919 


Io54o 


9 


48781 


I23c8 


8- 1 2481 


5 9 


2 


07051 


U 


•1821 


08807 


ii-354o 


IC569 


9 


46l4l 


12338 


8-io536 


58 


3 


07080 


14 


•1235 


08837 


ii-3i63 


io5 9 o 
10628 


9 


435i 5 


I236 7 


8 -08600 


57 


4 


07110 


14 


• o655 


08866 


11-2789 


9 


40904 


12397 


8-06674 


56 


5 


07189 


14 


•0079 


o88 9 5 


11-2417 


10657 


9 


38307 


12426 


8-04756 


55 


6 


07168 


1 3 


• 9507 


08925 


u-2048 


10687 


9 


35 7 24 


12456 


8-02848 


54 


7 


07197 


i3 


8940 


08954 


11-1681 


10716 


9 


33 1 54 


12485 


8 • 00948 


53 


8 


07227 


1 3 


• 83 7 8 


o8 9 83 


1 1 • i3i6 


10746 


9 


3o5q9 


I25i5 


7-99058 


5i 


9 


07206 


1 3 


•7821 


09013 


11 -0954 


io 77 5 


9 


•28o58 


12544 


7-97176 


5i 


10 


07285 


i3 


.7267 


09042 


11 -0594 
11 -0237 


io8o5 


9 


2553o 


12574 


7-95302 


30 


ii 


073l4 


1 3 


.6719 


09071 


io834 


9 


•23oi6 


I26o3 


7-93438 


% 


12 


07344 


1 3 


•6174 


09 1 01 


10-9882 


io863 


9 


2o5i6 


12633 


7-91082 


i3 


07373 


1 3 


5634 


09130 


10-9529 


10893 


9 


18028 


12662 


7-89734 


47 


14 


07402 


i3 


•5098 


09159 


10-9178 
10-8829 


10922 


9 


1 5554 


12692 


7 -8 7 8 9 5 


46 


i5 


0743l 


1 3 


4566 


09189 


10952 


9 


•13093 


12722 


7-86064 


43 


16 


07461 


i3 


4039 


09218 


io-8483 


10981 


9 


10646 


12751 


7-84242 


44 


17 


07490 


i3 


35i5 


09247 


10-8139 


lion 


9 


•08211 


12781 


7-82428 


43 


18 


075lo 


1 3 


2996 


09277 


10-7797 


1 1 040 


9 


•05789 


12810 


7-80622 


42 


19 


07548 


i3 


2480 


09306 


10-7437 


1 1 070 


9 


•03379 
00983 


12840 


7.78825 


4i 


20 


07578 


1 3 


1969 


09335 


10-7119 


1 1 099 




12869 


7-77035 


40 


21 


07607 


i3 


1461 


09365 


10-6783 


11128 


8 


98598 


12899 


7-75254 


\% 


22 


07636 


1 3 


og58 


09394 


io-645o 


in58 


8 


96227 


12929 


7.73480 


23 


07665 


i3 


0458 


09423 


10-6118 


11187 


8 


93867 


12958 


7-7HI5 


3i 


24 


07695 


12 


9962 


09453 


10-5789 


11217 


8 


9i52o 


12988 


7.69957 
7-68208 


36 


2 5 


07724 


12 


9469 


09482 


10-5462 


1 1 246 


8 


89185 


i3oi7 


35 


26 


07753 


12 


6981 


0951 1 


io-5i36 


11276 


8 


86862 


i3o47 


7 • 66466 


34 


27 


07782 


12 


8496 


09541 


io-48i3 


u3o5 


8 


8455i 


13076 


7-64732 


33 


28 


07812 


12 


8014 


09570 


10-4491 


n335 


S 


82252 


i3io6 


7-63oo5 


32 


2Q 


07841 


12 


7536 


09600 


10-4172 


u364 


8 


79964 


i3i36 


7.61287 


3i 


3o 


07870 


12 


7062 


09629 


10-3854 


1 1 394 


8 


77689 


i3i65 


7.59575 


3o 


3i 


07899 


12 


65 9 i 


og658 


io-3538 


II423 


8 


75425 


1 3 1 q5 


7.57872 


3 


32 


07929 


12 


6124 


09688 


10-3224 


1 1452 


8 


73172 


13224 


706176 


33 


079D8 


12 


566o 


09717 


10-2913 


1 1482 


8 


70931 


13254 


7-54487 


27 


34 


07987 


12 


5199 


09746 


10-2602 


1 i5i 1 


S 


68701 


13284 


7.52806 


26 


35 


08017 


12 


4742 


09776 


10-2294 


ii54i 


s 


66482 


i33i3 


7-5u32 


25 


36 


08046 


12 


4288 


09805 


10-1988 


1 1 570 


8 


64275 


13343 


7-49465 


24 


11 


08075 


12 


3838 


09834 


io-i683 


1 1 600 


8 


62078 


i33 7 2 


7-47806 


23 


08104 


12 


3390 


09864 


io-i38i 


11629 


8 


59893 


i34o2 


7-46i54 


22 


39 


o8i34 


12 


2946 


09893 


io- 1080 


1 1 65g 
1 1688 


8 


57718 


i343a 


7.44509 


21 


4o 


08 1 63 


12 


25o5 


09923 


10-0780 


8 


55555 


i346i 


7-42871 


20 


4i 


08192 


12 


2067 


09952 


1 • o483 


11718 


8 


53402 


1 349 1 


7.41240 


3 


42 


08221 


12 


i632 


09981 


10-0187 


11747 


8 


51259 


i352i 


7.39616 


43 


o825i 


12 


1201 


IOOII 


9-98930 


U777 


8 


49128 


i355o 


?J88? 


\l 


44 


08280 


12 


0772 


10040 


9-96007 


11S06 


8 


47007 


i35So 


45 


08309 


12 


o346 


10069 


9-93101 


1 1 836 


8 


44896 


1 3609 


7.34786 


13 


46 


o833 9 
o8368 


I I 


9923 


10099 
10128 


9-90211 


1 1 865 


8 


42795 


i363 9 


7.33190 


14 


47 


I I 


95o4 


9-87338 


1 1895 


8 


40705 


i366 9 
1 36 9 8 


7-3i6oo 


i3 


48 


08397 


I I 


9087 


ioi58 


9-84482 


1 1924 


8 


38625 


7-3ooi8 


12 


49 


08427 


I 1 


86 7 3 


10187 


9. 8 1 641 


1 1934 


8 


36555 


i3 7 28 


7-28442 


1 1 


5o 


08456 


I I 


8262 


10216 


9.78817 


11983 


8 


34496 


13758 


7 • 20873 


10 


5i 


08480 


I I 


7853 


10246 


9-76009 


I20l3 


8 


32446 


13787 


7-253io 


I 


52 


o85i4 


1 I 


7448 


10275 


9-73217 


12042 


8 


30406 


1 381.7 


7-23754 


53 


08544 


I I 


7045 


io3o5 


9-7o44i 


12072 


8 


28376 


i38i6 1 7-22204 


7 


54 


08573 


I I 


6645 


io334 


9-67680 


I210I 


8 


26355 


i38 7 6 


7-20661 


6 


55 


08602 


I 1 


6248 


io363 


9-64935 


I2l3l 


8 


24345 


13906 


7-19123 


5 


56 


o8632 


II. 


5853 


10393 


9-62205 


12160 


8 


22344 


i3g35 


7-17394 


4 


& 


08661 


II 


546i 


10422 


9 • 59490 


I 2 190 


8 


20352 


i3965 


7- 16071 


3 


08690 


I I 


5072 


io452 


9-56791 


12219 


S 


18370 


i3 99 5 


7- U553 


2 


59 


08720 


II 


4685 


10481 


9.54106 


12249 


8 


16398 


U024 


7>i3o42 


I 


6o 

/ 


08749 


ii-43oi 


IODIO 


9-5U36 


I2278 


8-14435 


Uo54 


7 • 1 1 537 


C 

/ 


CotJtng. 


Tangent. 


Cotang. 1 Tangent. 


Cotang. 


Tangent. 


Cotang. ■ Tangent. 


8 


5° 


84° 


83° 


82° 



Table III. NATURAL TANGENTS AND COTANGENTS. 76 


/ 


8° 


9° 


10° 


11° 


t 
60 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 





Uo54 


7 • 1 i537 


15838 


6.3i375 


17633 


5-67128 


I 9 438 


5-14455 


i 


14084 


7 


loo38 


15868 


6»3oi09 


17663 


5 


66i65 


19468 


5-i3658 


u 


2 


I4n3 


7 


o8546 


1 58 9 8 


6-29007 


17693 


5 


652o5 


19498 


5-12862 


3 


I4i43 


7 


070D9 


15928 


6-27829 


17723 


5 


64248 


19629 


5-12069 


57 


4 


I4I73 


7 


05579 


i5 9 58 


6-2665D 


I 77 53 


5 


63295 


19559 


5^11279 


56 


5 


14202 


7 


o4ioD 


1 5 9 88 


6-25486 


i 77 «3 


5 


62344 


19589 


5.1049c 


55 


6 


14232 


7 


02637 


1 60 1 7 


6-24321 


17813 


5 


6 1 397 


19619 


5-09704 


54 


7 


14262 


7 


01174 


16047 


6-23i6o 


17843 


5 


6o452 


19649 5-08921 


53 


8 


14291 


6 


99718 


16077 


6-22003 


17873 


5 


5951 1 


19680 


5>o8i39 


52 


9 


14321 


6 


98268 


16107 


6-2o85i 


17903 


5 


58573 


19710 


5.07360 


5i 


10 


U35i 


6 


96823 


i6i3 7 


6-19703 


17933 


5 


57638 


19740 


5-o6584 


5o 


ii 


i438i 


6 


95385 


16167 


6-18559 


17963 


5 


56706 


19770 


5-o58o9 


% 


12 


14410 


6 


93902 


161 96 


6-17419 


17993 


5 


55777 
5485i 


19801 


5-o5o37 


i3 


i444o 


6 


92325 


16226 


6-16283 


18023 


5 


1 983 1 


5-04267 


47 


14 


14470 


6 


91104 

89688 


16256 


6-i5i5i 


i8o53 


5 


53927 


19861 


5 -03499 


46 


i5 


14499 


6 


16286 


6-i4o23 


i8o83 


5 


53007 


1 989 1 


5-02734 


45 


16 


14529 


6 


88278 


i63i6 


6-12899 


i8n3 


5 


52090 


19921 


5-01971 


44 


17 


I455 9 
14588 


6 


86874 


16346 


6-11779 


18143 


5 


51176 


19952 


5-01210 


43 


18 


6 


85475 


■ 16376 


6- 10664 


18173 


5 


50264 


19982 


5 • oo45 1 


42 


*9 


14618 


6 


84082 


16405 


6-09552 


18203 


5 


4 9 356 


20012 


4.99695 


41 


20 


14648 


6 


82694 


16435 


6-o8444 


18233 


5 


4845i 


20042 


4-98940 


40 


21 


14678 


6 


8i3i2 


16465 


6-07340 


18263 


5 


47548 


20073 


4-98188 


% 


22 


14707 


6 


799 3 6 


16495 


6-06240 


18293 


5 


46648 


20103 


4-97438 


23 


14737 


6 


78064 


i6525 


6-o5i43 


i83 2 3 


5 


4575i 


?oi33 


4 • 96690 


37 


24 


14767 


6 


77199 


i6555 


6-o4o5i 


i8353 


5 


4485 7 


20164 


4-95945 


36 


25 


1 479° 


6 


75838 


16585 


6-02962 
6-01878 


1 8383 


5 


43966 


20194 


4.95201 


35 


26 


14826 


6 


74483 


i66i5 


18414 


5 


43077 


20224 


4.94460 


34 


27 


14856 


6 


73i33 


16645 


6-00797 


18444 


5 


42192 


20254 


4.93721 


33 


28 


14886 


6 


71789 


16674 


5-99720 


18474 


5 


4i3o9 


20285 


4-92984 


32 


2 9 


I49I3 


6 


7o45o 


16704 


5.98646 


i85o4 


5 


40429 


2o3i5 


4-92249 


3i 


3o 


14945 


6 


69116 


16734 


5-97576 


18534 


5 


39552 


2o345 


4"9i5i6 


3o 


3i 


14975 


6 


67787 


16764 


5-965io 


1 8564 


5 


38677 


20376 


4.90785 


% 


32 


i5oo5 


6 


66463 


16794 


5.95448 


i85 9 4 


5 


37805 


20406 


4-90056 
4-8933o 


33 


i5o34 


6 


65 1 44 


16824 


5-94390 
5.93335 


18624 


5 


36 9 36 


20436 


27 


34 


i5o64 


6 


6383 1 


i6854 


i8654 


5 


36070 


20466 


4-886o5 


26 


35 


i5o 9 4 


6 


62523 


16884 


5.92283 


18684 


5 


35206 


20497 


4-87882 


25 


36 


i5i24 


6 


61219 


16914 


5.91235 


18714 


5 


34345 


20327 


4-87162 


24 


37 


i5i53 


6 


59921 


16944 


5.90191 
5.8 9 i5i 


18745 


5 


33487 


20557 


4-86444 


23 


38 


i5i83 


6 


58627 


16974 


i8 77 5 


5 


3263 1 


2o588 


4-85727 


22 


3 9 


i5 2 i3 


6 


57339 


17004 


5-88114 


i88o5 


5 


3i 77 8 


20618 


4-85oi3 


21 


4o 


15243 


6 


56o55 


17033 


5-87080 


18835 


5 


30928 


20648 


4-843oo 


20 


4i 


15272 


6 


54777 


17063 


5-86o5i 


18865 


5 


3oo8o 


20679 


4.83590 


I? 


42 


i53o2 


6 


535o3 


17093 


5-85o24 


i88 9 5 


5 


29235 


20709 


4-82882 


43 


15332 


6 


52234 


17123 


5-84ooi 


18925 


5 


283 9 3 


20739 


4-82175 


u 


44 


1 5362 


6 


50970 


I7i53 


5-82982 


i8 9 55 


5 


27553 


20770 


4-81471 


45 


i53gi 


6 


497 I0 


17183 


5-81966 


18986 


5 


26715 


20800 


4-80769 


i5 


46 


1 542 1 


6 


48456 


17213 


5-80953 


19016 


5 


2588o 


2o83o 


4 -80068 


14 


% 


1 545 1 


6 


47206 


17243 


5-79944 


19046 


5 


25o48 


20861 


4-7937° 


i3 


1 548 1 


6 


45961 


17273 


5- 7 8 9 38 


19076 


5 


24218 


20891 


4.78673 


12 


i 9 


i55ii 


6 


44720 


i73o3 


5- 779 36 


19106 


5 


23391 


20921 


4.77978 


1 1 


5o 


1 554o 


6 


43484 


i 7 333 


5.76937 


19136 


5 


22566 


20952 


4-77286 


10 


5i 


15570 


6 


42253 


17363 


5.75941 


19166 


5 


21744 


20982 


4-76395 


9 


52 


i56oo 


6 


41026 


i73 9 3 


5-74949 


19197 


5 


20925 


2IOl3 


4-75906 


8 


53 


i563o 


6 


3 9 8o4 

3858 7 


17423 


5-73960 


19227 


5 


20107 


21043 


4-/52I9 


7 


54 


i566o 


6 


17453 


5.72974 


i 9 25 7 


5 


19293 
18480 


21073 


4-74534 


6 


55 


i568 9 


6 


37374 


i 7 483 


5-71992 


19287 


5 


21 104 


4-7385i 


5 


56 


15719 


6 


36i65 


i 7 5i3 


5-7ioi3 


19317 


5 


1 767 1 


2ii34 


4-73i70 


4 


u 


15749 


6 


34961 


17543 


5-70037 


19347 


5 


1 6863 


21 164 


4-72490 


3 


. 15779 


6 


33 7 6i 


17573 


5 • 69064 


i 9 3 7 8 


5 


i6o58 


21195 


4-7i8i3 


2 


5 9 


1 58o 9 
1 5838 


6 


32566 


17603 


5-68094 


19408 


5 


i5256 


21225 


4-7ii37 
4-70403 


1 


6o 


6-3i3 7 5 


i 7 633 


5-67128 


i 9 438 


5-14455 


21256 





/ 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


1 


81° 


80° 


79° 


78° 



76 NATURAL TANGENTS AND COTANGENTS. Table IIL 


/ 


12° 


13° 


14° 


15° 


/ 

60 


Tangent. 


Cotang. 1 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 





21256 


4-70463 
4-69791 


23o87 


4-33U8 


24933 


4-01078 
4-oo582 


26795 


3-732o5 


i 


21286 


23i 17 


4-32573 


24964 


26826 


3.72771 
3-72338 


U 


2 


2i3i6 


4-69121 


23 148 


4-32ooi 


24995 


4 -00086 


26857 


3 


2 1 347 


4-68452 ! 


23179 


4-3i43o 


25026 


3-99592 


26888 


3.71907 


5? 


4 


2i377 
21408 


4-67786 


23209 


4-3o86o 


25o56 


3.99099 


26920 


3-71476 


56 


5 


4-67121 


23240 


4-30291 


25087 


3-98607 


26951 


3-71046 


55 


6 


21438 


4-66458 


23271 


4-29724 


25u8 


3-98117 


26982 


3-70616 


54 


7 


21469 


4-65797 
4-65i38 


233oi 


4-29159 
4-28593 
4- 28032 


25 1 49 


3-97627 


270l3 


3-70188 
3-69761 


53 


u 


21499 


23332 


25i8o 


3-97139 


27044 


52 


9 


21529 


4-64480 


23363 


252II 


3- 9 665 1 


27076 


3.69335 
3 • 68909 


5i 


10 


2i56o 


4-63825 


23393 


4-27471 


25242 


3.96165 


27107 


5o 


li 2i5go 


4-63i7i 


23424 


4-26911 
4-26352 


25273 


3-9568o 


27i38 


3-68485 


% 


12 


21621 


4-625i8 


23455 


2 53 04 


3-95196 


27169 


3 -68061 


i3 


2i65i 


4-6i868 


23485 


4-25795 


25335 


3-94713 


27201 


3-67638 


47 


14 


21682 


4-61219 


2 35i6 


4-2523o 
4-24683 


25366 


3-94232 


27232 


3-67217 


46 


i5 


21712 


4-60072 


23547 


25397 


3-93751 


27263 


3-66796 


45 


16 


21743 


4-59927 


2 35 7 8 


4-24i32 


25428 


3-93271 


27294 


3-663 7 6 


44 


17 


21773 


4- 5 9 283 


236o8 


4-2358o 


25459 


3-92793 


27326 


3>65g57 
3-65538 


43 


18 


21804 


4-58641 


23639 


4-23o3o 


25490 


3-923i6 


27357 


42 


19 


2i834 


4-58ooi 


23670 


4-22481 


25521 


3- 9 i83 9 


2 7 388 


3-65i2i 


4i 


20 


21864 


4-57363 


23700 


4-21933 


25552 


3 -91 364 


27419 


3 -647o5 


4o 


21 


21895 


4-56726 


2373i 


4-21387 


25583 


3-90890 


2745i 


3-64289 


ll 


22 


21925 


4-56091 


23762 


4-20842 


256i4 


3-90417 


27482 


3-638 7 4 


23 


21956 


4-55438 


23793 


4-20298 


25645 


3-89945 


27613 


3-63461 


37 


34 


21986 


4-54826 


23823 


4-19736 


25676 


3-89474 


27545 


3-63o48 


36 


25 


22017 


4-54196 


23854 


4- io2i5 


25707 


3-89004 


27576 


3-62636 


35 


26 


22047 


4-53568 


23885 


4- i86 7 5 


2 5 7 38 


3-88536 


27607 
07638 


3-62224 


34 


27 


22078 


4-52941 
4-523i6 


23916 


4-18137 


25769 


3 -88068 


3-6i8u 


33 


28 


22108 


23946 


4-17600 


258oo 


3-87601 


27670 


3-6i4o5 


32 


? 9 


22139 


4-5i6g3 


23977 


4-17064 


2583i 


3-87136 


27701 


3-60996 


3i 


3o 


22169 


4*51071 


24008 


4-i653o 


25862 


3-86671 


2 77 32 


3-60388 


3o 


3i 


22200 


4-5o45i 


24039 


4-i5o97 


2 58 9 3 


3-86208 


27764 


3-60181 


3 


32 


22231 


4-49832 


24069 


4- 1 5465 


25924 


3-85745 


27793 


3 -59775 


33 


22261 


4-4o2i5 


24100 


4-I4934 


25955 


3-85284 


27826 


3-59370 


27 


34 


22292 


4-48600 


24i3i 


4-U4o5 


25 9 86 


3-84824 


2 7 858 


3 • 58966 


26 


35 


22322 


4-47986 


24162 


4-i38 77 


26017 
26048 


3-84364 


27889 


3-58562 


25 


36 


22353 


4-47374 


24193 


4-i335o 


3-83 9 o6 


27920 


3-58i6o 


24 


37 


22383 


4-46764 


24223 


4-I2825 


26079 


3-83449 


27932 


3-57758 


23 


38 


22414 


4-46i55 


24254 


4-i23oi 


26110 


3-82992 


27983 


3-57357 


22 


39 


22444 


4-45548 


24285 


4-11778 


26141 


3-82537 


28oi5 


3-56957 
3- 5655 7 


21 


40 


22475 


4-44942 
4-44338 


243 1 6 


4-U256 


26172 


3-82083 


28046 


20 


41 


225o5 


24347 


4-10736 


26203 


3-8i63o 


28077 


3-56i5 9 


3 


42 


22536 


4-43735 


24377 


4- 10216 


26235 


3.81177 


28109 


3-35761 


43 


22567 


4-43i34 


24408 


4 • 09699 


26266 


3-80726 


28140 


3-55364 


n 


44 


22597 


4-42534 


24439 


4-09182 
4 -08666 


26297 


3-80276 


28172 


3-54968 


16 


45 


22628 


4-4I936 


24470 


26328 


3.79827 


28203 


3.54673 


i5 


46 


22658 


4-4i34o 


245oi 


4-o8i52 


26359 


3.79378 


28234 


3-54H9 


14 


2 


22689 


4-40745 


24532 


4-07639 


26390 


3-78931 


28266 


3-53 7 85 


i3 


22719 


4-4oi52 


24562 


4-07127 


26421 


3-78485 


28297 


3-53393 


12 


i 9 


22750 


4-39560 
4-38969 
4-3838i 


245g3 


4-06616 


26452 


3 ■ 78040 


2S32 9 


3-53ooi 


11 


5o 


22781 


24624 


4-06107 


26483 


3-77595 


2836o 


3-52609 


10 


5i 


22811 


24655 


4- 05599 


265i5 


3.77102 


283 9 i 


3-52219 


2 


52 


22842 


4-37793 


24686 


4-05092 
4-04586 


26546 


3.76709 


28423 


3-51829 


53 


22872 


4-37207 


24717 


26577 


3.76268 


28454 


3-5i44i 


I 


54 


22903 


4-36623 


24747 


4 -04081 


26608 


3.75828 


2S4S6 


3-5io53 


55 


22934 


4-36o4o 


24778 


4-o3578 


26639 


3-75388 


28517 


3 • 5o666 


5 


56 


22964 


4-35459 


24809 


4-o3o75 


26670 


3.74950 


28549 


3- 50279 


4 


n 


22995 


4-34879 


24840 


4-02574 


26701 


3.74312 


285So 


3-49894 


3 


23026 


4-343oo 


24871 


4 02074 


26733 


3-74075 


28612 


3-495c>9 


2 


5 9 


23o56 


4-33723 


24902 


4-01576 


26764 


3«7364o 


28643 


3-49125 


I 


60 


23087 


4-33U8 


24933 


4*01078 


26795 


3«732o5 


28675 


3-48741 


O 

/ 


/ 


Cotang. | Tangent. 


Cotang. Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 




T?° 


753 


75° 


14° 



Table IIL NATURAL TANGENTS AND COTANGENTS. 77 


i 


16° 


11° 


18° 


19° 


/ 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 





28675 


3-4874I 


3o573 


3-27085 


32492 


3.07768 


34433 


2-9042I 


60 


i 


28706 


3 


48359 


3o6o5 


3-26745 


32524 


3 


07464 


34465 


2 


90147 


ss 


2 


28 7 38 


3 


47977 


3o637 


3 


26406 


32556 


3 


07160 


34498 


2 


89873 


3 


28769 


3 


47096 


30669 


3 


26067 


32588 


3 


o685 7 


3453o 


2 


89600 


sz 


4 


28800 


3 


47216 


30700 


3 


25729 


32621 


3 


o6554 


34563 


2 


8 9 32 7 


5 


28832 


3 


46837 


30732 


3 


253o2 
25o55 


32653 


3 


06252 


34596 


2 


8 9 o53 


55 


6 


28864 


3 


46458 


30764 


3 


32685 : 


o5g5o 


34628 


2 


88 7 83 


54 


7 


28895 


3 


46080 


30796 | 3 


24719 

24383 


32717 


3 


05649 


3466i 


2 


885u 


53 


8 


28927 
289D8 


3 


45703 


30828 


3 


32749 


3 


o5349 


34693 


2 


88240 


52 


9 


3 


45327 


3o86o 


3 


24049 


32782 


3 


o5o49 


34726 


2 


87970 


5i 


10 


28990 


3 


44931 


30891 


3 


23714 


32814 


3 


04749 


34758 


2 


87700 


5o 


ii 


29021 


3 


44376 


30921 


3 


2338i 


32846 


3 


o445o 


34791 


2 


87430 


% 


12 


2Q053 


3 


44202 


30955 


3 


23o48 


32878 


3 


o4i52 


34824 


2 


87161 


i3 


29084 


3 


43829 


30987 


3 


22715 


32911 


3 


o3854 


34856 


2 


86892 


47 


i4 


29116 


3 


43456 


31019 


3 


22384 


32943 


3 


o3556 


34889 


2 


86624 


46 


i5 


29147 


3 


43o84 


3io5i 


3 


22053 


32975 


3 


o326o 


34922 


2 


86356 


45 


16 


29179 


3 


42713 


3io83 


3 


21722 


33007 


3 


02963 


34954 


2 


86089 


44 


n 


29210 


3 


42343 


3 1 1 1 5 


3 


21392 


33o4o 


3 


02667 


34987 


2 


85822 


43 


18 


29242 


3 


41973 


3 1 147 


3 


2io63 


33072 


3 


02372 


35019 


2 


85555 


42 


19 


29274 


3 


41604 


3n 7 8 


3 


20734 


33 1 04 


3 


02077 


35o52 


2 


8528 9 
85o23 


41 


20 


293o5 


3 


41236 


3l2IO 


3 


20406 


33 1 36 


3 


01783 


35o85 


2 


40 


21 


29337 
29368 


3 


40869 


3l242 


3 


20079 


33169 


3 


01489 


S5ii 7 


2 


84758 


% 


22 


3 


4o5o2 


31274 


3 


19752 


33201 


3 


01 196 


35i5o 


2 


84494 


23 


29400 


3 


40 1 36 


3i3o6 


3 


19426 


33233 


3 


00903 


35i83 


2 


84229 
83 9 65 


37 


24 


29432 


3 


39771 


3i338 


3 


19100 


33266 


3 


0061 1 


352i6 


2 


36 


25 


29463 


3 


39406 


3i37o 


3 


18775 


33298 


3 


00319 


35248 


2 


83702 


35 


26 


29495 


3 


39042 


3 1 402 


3 


1 845 1 


3333o 


3 


00028 


3528i 


2 


83439 


34 


27 


29526 


3 


38679 


3 1 434 


3 


18,27 


33363 


2 


99738 


353i4 


2 


83176 


33 


28 


2 9 558 


3 


383i 7 


31466 


3 


17804 


33395 


2 


99447 


35346 


2 


82914 


3a 


29 


29390 


3 


37955 


31498 


3 


I748i 


33427 


2 


991 58 


35379 


2 


82653 


3i 


3o 


29621 


3 


37094 


3i53o 


3 


17159 


3346o 


2 


98868 


35412 


2 


823 9 i 


3o 


3i 


2 9 653 


3 


37234 


3i562 


3 


16838 


33492 


2 


9858o 


35445 


2 


82i3o 


11 


32 


29685 


3 


36875 


3 1594 


3 


i65i 7 


33524 


2 


98292 


35477 


2 


81870 


33 


29716 


3 


365i6 


3i6 2 6 


3 


16197 


33557 


2 


98004 


355io 


2 


81610 


11 


34 


29748 


3 


36i58 


3 1 658 


3 


1 58 77 


3358 9 


2 


97717 


35543 


2 


8i35o 


35 


29780 


3 


358oo 


31690 


3 


15558 


33621 


2 


9743o 


355 7 6 


2 


81091 


25 


36 


2981 1 


3 


35443 


31722 


3 


1 524o 


33654 


2 


97144 


356o8 


2 


8o833 


24 


\l 


29843 


3 


35o8 7 


3i754 


3 


14922 


33686 


2 


9 6858 


35641 


2 


80574 


23 


38 


29875 


3 


34732 


3i 7 86 


3 


i46o5 


33718 


2 


96573 


35674 


2 


8o3i6 


22 


3 9 


29906 


3 


34377 


3i8i8 


3 


14288 


33 7 5i 


2 


96288 


35707 


2 


80059 


21 


4o 


29938 


3 


34023 


3i85o 


3 


13972 


33783 
338i6 


2 


96004 


35740 


2 


79802 


20 


4i 


29970 


3 


33670 


3i882 


3 


13656 


2 


9 5 7 2I 


33772 


2 


79545 


\l 


42 


3oooi 


3 


333i 7 
32 9 65 


31914 


3 


i334i 


33848 


2 


95437 


358o5 


2 


.79289 


43 


3oo33 


3 


31946 


3 


13027 


3388i 


2 


9 5i55 


35838 


2 


.79033 


17 


44 


3oo65 


3 


32614 


31978 


3 


12713 


339i3 


2 


94872 


358 7 i 


2 


78778 


16 


45 


30097 


3 


32264 


32010 


3 


12400 


33945 


2 


94590 


33904 


2 


•78523 


i5 


46 


3oi28 


3 


3ioi4 

3 1 565 


32042 


3 


12087 


33978 


2 


94309 


35 9 3 7 


2 


.78269 


14 


% 


3oi6o 


3 


32074 


3 


11775 


34oio 


2 


94028 


35969 


2 


.78014 


i3 


30192 


3 


3i2i6 


32106 


3 


1 1 464 


34043 


2 


93748 


36oo2 


2 


.77761 


12 


49 


30224 


3 


3o868 


32 139 


3 


1 1 1 53 


34075 


2 


93468 


36o35 


2 


•77507 


1 1 


5o 


3o255 


3 


30321 


32171 


3 


10842 


34io8 


2 


93189 


36o68 


2 


•77254 


10 


5i 


30287 


3 


30174 


32203 


3 


io532 


34i4o 


2 


92910 


36ioi 


2 


•77002 


i 


52 


3o3i9 


3 


29829 
2 9 483 


32235 


3 


10223 


34h3 


2 


92632 


36i34 


2 


•76750 


53 


3o35i 


3 


32267 


3 


09914 


342o5 


2 


92354 


36i6 7 


2 


76498 


7 


54 


3o382 


3 


2Ql39 


32299 


3 


09606 


34238 


2 


92076 


36199 


2 


•76247 


6 


55 


3o4U 


3 


28793 


3233i 


3 


09298 


34270 


2 


9*799 
9i523 


36 2 32 


2 


.75996 


5 


56 


3o446 


3 


28452 


32363 


3 


08901 


343o3 


2 


36 2 65 


2 


•75746 


4 


n 


30478 


3 


28109 


32396 


3 


08685 


34335 


2 


91246 


36298 
3633 1 


2 


•75496 


3 


3o5o9 


3 


27767 


32428 


3 


08379 


34368 


2 


90971 


2 


•75246 


2 


5 9 


3o54i 


3 


•27426 


3246o 


3 


08073 


344oo 


2-90696 


36364 


2 


74997 


1 


60 


3o573 


3.27085 


32492 


3-07768 


34433 


2.90421 


36397 


2.74748 





# 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


1 


73° 


72° 


71° 


70° 



78 NATURAL TANGENTS AND COTANGENTS. Table III. 


/ 


20° 


21° 


22° 


23° 


60 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 





363g7 
3643o 


2-74748 


38386 


2»6o5o9 


4o4o3 


2-47509 


42447 


2-35585 


i 


2 


74499 


38420 


2 


60283 


4o436 


2 


473o2 


42482 


2 


•35390 


u 


2 


36463 


2 


7425i 


38453 


2 


6oo57 


40470 


2 


47095 


425i6 


2 


■352o5 


3 


36496 


2 


74oo4 


38487 


2 


5 9 83i 


4o5o4 


2 


46888 


4255i 


2 


•35oi5 


57 


4 


36529 


2 


73756 


38520 


2 


59606 


4o538 


2 


46682 


42585 


2 


•34825 


56 


5 


36562 


2 


735co 


38553 


2 


5 9 38i 


40572 


2 


46476 


42619 


2 


-34636 


55 


6 


365g5 


2 


73263 


38587 


2 


5 9 i56 


40606 


2 


46270 


42654 


2 


•34447 


54 


I 


36628 


2 


73017 


38620 


2 


58932 


40640 


2 


46o65 


42688 


2 


•34258 


53 


3666i 


2 


72771 


38654 


2 


58708 


40674 


2 


4586o 


42722 


2 


•34069 


52 


9 


36694 


2 


72526 


38687 


2 


58484 


40707 


2 


45655 


42757 


2 


•3388i 


5i 


10 


36727 


2 


72281 


38721 


2 


58 2 6i 


40741 


2 


4545i 


42791 


2 


336 9 3 


5o 


u 


36760 


2 


72o36 


38 7 54 


2 


58o38 


40775 


2 


45246 


42826 


2 


335o5 


% 


12 


36793 


2 


71792 


38787 


2 


5 7 8 1 5 


40809 
40843 


2 


45o43 


42860 


2 


•333i 7 


i3 


36826 


2 


7 i548 


38821 


2 


57593 


2 


4483 9 


42894 


2 


•33i3o 


47 


i4 


3685 9 


2 


7i3o5 


38854 


2 


57371 


40877 


2 


44636 


42929 


2 


•32943 


46 


i-5 


36892 


2 


71062 


38888 


2 


57i5o 


409 1 1 


2 


44433 


42963 


2 


•32756 


45 


16 


36925 


2 


70819 


38 9 2i 


2 


56928 


40945 


2 


442 3 


42998 


2 


•32570 


44 


17 


36 9 58 


2 


7o5 77 


38955 


2 


56707 


40979 


2 


44027 


43o32 


2 


•32383 


43 


i.8 


36991 


2 


7o335 


38 9 88 


2 


56487 


4ioi3 


2 


43825 


43067 


2 


.32197 


42 


19 


37024 


2 


70094 
6 9 853 


39022 


2 


56266 


41047 


2 


43623 


43ioi 


2 


■32012 


41 


20 


37057 


2 


39055 


2 


56o46 


41081 


2 


43422 


43 1 36 


2 


.31826 


40 


21 


37090 


2 


69612 


39089 


2 


55827 


4iu5 


2 


43220 


43170 


2 


•3i64i 


M 


22 


37124 


2 


69371 


39122 


2 


556o8 


41U9 


2 


43oig 


432o5 


2 


•3i456 


23 


37i5 7 


2 


6913 1 
68892 


3 9 i56 


2 


5538 9 


4ii83 


2 


42819 


43239 


2 


•31271 


37 


24 


37190 


2 


39190 


2 


55i7o 


41217 


2 


42618 


43274 


2 


3 1 086 


36 


25 


37223 


2 


68653 


39223 


2 


54952 


4i25i 


2 


42418 


433o8 


2 


•30902 


35 


26 


37256 


2 


68414 


39257 


2 


54734 


41285 


2 


42218 


43343 


2 


•30718 


34 


27 


37289 


2 


68175 


39290 


2 


545i6 


4i3i9 
4i353 


2 


42019 


43378 


2 


3o534 


33 


28 


37322 


2 


67937 


39324 


2 


54299 


2 


41819 


43412 


2 


3o35i 


32 


29 


3 7 355 


2 


67700 


39357 


2 


54082 


4i387 


2 


41620 


43447 


2 


30167 


3i 


3o 


3 7 388 


2 


67462 


39391 


2 


53865 


41421 


2 


4U2I 


4348i 


2 


29984 


3o 


3i 


37422 


2 


67225 


3 9 425 


2 


53648 


4i455 


2 


41223 


435 1 6 


2 


29801 


ll 


32 


37455 


2 


66989 


3 9 458 


2 


53432 


41490 


2 


41025 


4355o 


2 


29619 


33 


37488 


2 


66752 


39492 


2 


53217 


4i524 


2 


40827 


43585 


2 


29437 


27 


34 


37521 


2 


665 1 6 


39526 


2 


53ooi 


4i558 


2 


40629 


43620 


2 


29204 


26 


35 


37554 


2 


66281 


39559 
39593 


2 


52786 


41592 


2 


4o432 


43654 


2 


29073 


25 


36 


37588 


2 


66046 


2 


52571 


41626 


2 


40235 


4368 9 


2 


28891 


24 


ll 


37621 


2 


658u 


39626 


2 


5 2 35 7 


41660 


2 


4oo38 


43724 


2 


28710 


23 


37654 


2 


65576 


39660 


2 


52142 


41694 


2 


3 9 84i 


43758 


: 


28528 


22 


3 9 


37687 


2 


65342 


39694 


2 


51929 


41728 


2 


39645 


43 79 3 


2 


28348 


21 


4o 


37720 


2 


65109 


39727 


2 


5i7i5 


41763 


2 


39449 


43828 


2 


28167 


20 


4i 


37754 


2 


64875 


39761 


2 


5i5o2 


41797 


2 


3 9 253 


43862 


2 


27987 
27806 


19 


42 


37787 
37820 


2 


64642 


39795 


2 


51289 


4i83i 


2 


3 9 o58 


438 97 


2 


18 


43 


2 


64410 


39829 


2 


51076 


4 1 865 


2 


38862 


43932 


2 


27626 


17 


44 


37853 


2 


64177 


39862 


2 


5o864 


41899 


2 


38668 


43 9 66 


2 


27447 


16 


45 


37887 


2 


63 9 45 


39896 


2 


5o652 


41933 


2 


38473 


44001 


2 


27267 


i5 


46 


37920 


2 


63 7 i4 


39930 


2 


5o44o 


41968 


2 


38279 


44o36 


2 


27088 


14 


47 


37953 


2 


63483 


39963 


2 


50229 


42002 


2 


38o84 


44071 


2 


26909 


i3 


48 


3 79 86 


2 


63252 


39997 


2 


5ooi8 


42036 


2 


37891 


44io5 


2 


26730 


12 


49 


38o2o 


2 


63o2i 


4oo3i 


2 


49807 


42070 


2 


37697 


44i4o 


2 


26552 


11 


5o 


38o53 


2 


62791 


4oo65 


2 


49:597 
49386 


42IOD 


2 


375o4 


44n5 


2 


263 7 4 


10 


5i 


38o86 


2 


62D61 


40098 


2 


42i3o 


2 


373u 


442io 


2 


26196 


I 


52 


38i2o 


2 


62332 


4oi32 


2 


4QI77 


42173 


2 


37118 


44244 


2 


26018 


53 


38i53 


2 


62io3 


40166 


2 


48967 


42207 


2 


36925 


44279 


2 


25840 


7 


54 


38 1 86 


2 


61874 


40200 


2 


48 7 58 


42242 


2 


36 7 33 


443i4 


2 


25663 


6 


55 


38220 


2 


61646 


40234 


2 


48549 


42276 


2 


36541 


44349 


2 


25486 


5 


56 


38253 


2 


61418 


40267 


2 


48340 


423 10 


2 


36349 


44384 


2 


25309 


4 


57 


38286 


2 


61190 


4o3oi 


2 


48i32 


42345 


2 


36i58 


444i8 


2 


25l32 


3 


58 


38320 


2 


60963 


4o335 


2 


47924 


42370 
42413 


2 


35967 


44453 


2 


24956 


2 


5 9 


38353 


2 


60736 


4o36o 
4040J 


2 


477i6 


2 


35776 


444S8 


2 


24780 


1 


60 


38*386 


2 '60509 


2-47509 


42447 


2-35585 


44523 


2 • 24604 





Cotang. 


Tangent. ' 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


/ 




69° 


68° 


67° 


66° 



Table ITL NATURAL TANGENTS AND COTANGENTS. 79 


t 




24° 


25° 


26° 


27° 


/ 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. C 


'otang. 


44523 


2 • 24604 


4663 1 


2-i445i 


48773 


2-o5o3o 


5o 9 53 1 


96261 


60 


i 


44558 


2 


•24428 


46666 


2 


14288 


48809 


2 


04879 


50989 I 


96120 


It 


2 


445 9 3 


2 


•24252 


46702 


2 


I4i25 


48845 


2 


04728 


61026 1 


95979 

9 5838 


3 


44627 


2 


•24077 


46737 


2 


1 3 9 63 
i38oi 


48881 


2 


04577 


5io63 1 


57 


4 


44662 


2 


•23902 


46772 


2 


48917 


2 


•04426 


51099 1 


95698 


56 


5 


44697 
44732 


2 


•23 7 27 


46808 


2 


1 3639 


48953 


2 


04276 


5n36 1 


95537 


55 


6 


2 


23553 


46843 


2 


13477 


48989 


2 


•o4i25 


5 1 173 1 


95417 


54 


7 


44767 
44802 


2 


23378 


46879 


2 


i33i6 


49026 


2 


03975 


51209 1 


95277 


53 


8 


2 


23204 


46914 


2 


i3i54 


49062 


2 


•o3825 


5i246 1 


93137 


52 


9 


4483 7 


2 


23o3o 


46950 


2 


:%! 


49098 


2 


03675 


5i283 1 


94997 


5i 


IO 


44872 


2 


2285 7 


46985 


2 


49134 


2 


•o3526 


5i3i 9 1 


94838 


5o 


ii 


44907 


2 


22683 


47021 


2 


1 267 1 


49170 


2 


•03376 


5i356 1 


94718 


% 


12 


44942 


2 


225lO 


47c 56 


2 


I25ll 


49206 


2 


•03227 


5 1 393 1 


94579 


i3 


44977 


2 


22337 


47092 


2 


i235o 


49242 


2 


•03078 


5i43o 1 


9444o 


% 


14 


45oi2 


2 


22164 


47128 


2 


1 2 100 
i2o3o 


49278 


2 


•02929 


51467 1 


943oi 


i5 


45o47 


2 


21992 


47163 


2 


493 1 5 


2 


•02780 


5i5o3 1 


94162 


45 


16 


45o82 


2 


21819 


47199 


2 


11871 


4935i 


2 


•0263 1 


5 1 54o 1 


94023 


44 


n 


45li7 


2 


21647 


47234 


2 


11711 


49387 


2 


•02483 


5i5 77 1 


9 3885 


43 


18 


45i52 


2 


21475 


47270 


2 


■n552 


49423 


2 


•02335 


5i6i4 1 


93746 


42 


19 


45187 


2 


2i3o4 


473o5 


2 


1 1392 
H233 


4945^ 


o 


•02187 


5i65i I 


936o8 


41 


20 


45222 


2 


2JI32 


47341 


2 


49493 


2 


02039 


5i688 1 


93470 


40 


21 


45 2 07 


J 


20961 


47377 


2 


11075 


49532 


2 


•01891 


51724 1 


93332 


ll 


22 


45292 


2 


20790 


47412 


2 


10916 


49568 


2 


01743 


61761 1 


93195 
93067 


23 


45327 


2 


20619 


47448 


2 


io 7 58 


49604 


2 


•01596 


5i 79 8 1 


37 


24 


45362 


2 


20449 


47483 


2 


10600 


49640 


2 


01449 


5i835 1 


92920 


36 


25 


45397 
45432 


2 


20278 


47519 


2 


10442 


49677 
497i3 


2 


Ol3o2 


51872 1 


92782 


35 


26 


2 


20108 


47555 


2 


10284 


2 


ou55 


51909 1 


92645 


34 


27 


45467 


2 


19938 


47590 


2 


10126 


49749 


2 


01008 


51946 1 


925o8 


33 


28 


455o2 


2 


19769 


47626 


2 


09969 


49786 


2 


00862 


5i 9 83 1 


92371 


32 


2 9 


45537 


2 


19399 

19430 


47662 


2 


0981 1 


49822 


2 


00715 


52020 I 


92235 


3i 


3o 


45573 


2 


47698 


2 


09654 


49858 


2 


00569 


52057 I 


92098 


3o 


3i 


456o8 


2 


19261 


47733 


2 


09498 


49894 
49931 


2 


oo423 


52094 I 


91962 


2 


32 


456/ + 3 


2 


19092 

18923 


47769 
47805 


2 


09341 


2 


00277 


52 i 3 i 1 


91826 


33 


456 7 8 


2 


2 


09184 


49967 


2 


ooi3i 


52i68 1 


91690 


27 


34 


457i3 


2 


18755 


47840 


2 


09028 


5ooo4 


1 


90986 


52205 I 


9i554 


26 


35 


45748 


2 


i858 7 


47876 


2 


08872 


5oo4o 


I 


99841 


52242 I 


91418 


25 


36 


45784 


2 


18419 


47912 


2 


08716 


50076 


I 


99695 
99560 


52279 ! 


91282 


24 


37 


43819 


2 


i8 2 5i 


47948 


2 


o856o 


5on3 


I 


523i6 1 


9H47 


23 


38 


45854 


2 


18084 


47984 


2 


o84o5 


5oi49 
5oi85 


I 


99406 


52353 1 


91012 


22 


3 9 


40889 


2 


17916 


48019 


2 


o825o 


I 


99261 


52390 1 


90876 


21 


4o 


45924 


2 


17749 


48o55 


2 


08094 
07939 

07786 


50222 


I 


991 16 


52427 1 


90741 


20 


4i 


45960 


2 


17582 


48091 


2 


5o258 


1 


98972 
98828 


52464 1 


90607 


\t 


42 


45995 


2 


17416 


48127 


2 


50295 


I 


525oi 1 


90472 


43 


46o3o 


2 


17249 


48i63 


2 


07630 


5o33i 


I 


98684 


52538 1 


90337 


17 


44 


46o65 


2 


17083 


48198 


2 


07476 


5o368 


I 


98540 


52575 1 


90203 


16 


45 


46101 


2 


16917 


48234 


2 


07321 


5o4o4 


I 


9 83 9 6 


526i3 1 


90069 


i5 


46 


46 1 36 


2- 


16751 


48270 


2 


07167 


5o44i 


I 


98253 


5265o 1 


89935 


14 


% 


46171 


2 


16585 


483o6 


2 


07014 


5o477 


I 


98110 


5 2 68 7 1 


89801 


i3 


46106 


2 


16420 


48342 


2 


06860 


5o5i4 


I 


97966 


52724 1 


89667 


12 


P 


46242 


2 


16255 


48378 


2 


06706 


5o55o 


I 


97823 


52761 1 


8 9 533 


n 


5o 


46277 


2 


16090 


48414 


2 


o6553 


5o58 7 


I 


97680 


52708 1 


89400 


10 


5i 


463 1 2 


2 


15925 


4845o 


2 


06400 


5o623 


I 


97538 


52836 1 


89266 


8 


52 


46348 


2 


15760 


48486 


2 


06247 


5o66o 


I 


97395 


52873 1 


8 9 i33 


53 


46383 


2 


i55q6 


48521 


2 


06094 


50696 
5o733 


I 


972D3 


52910 1 


89000 


7 


54 


46418 


2 


1 543 2 


48557 


2- 


05942 


1 


97m 


52947 1 


88867 


6 


55 


46454 


2 


1 5268 


485g3 


2- 


05790 


50769 


I 


96969 


52984 1 


88734 


5 


56 


46489 


2 


i5io4 


48629 
48665 


2- 


05637 


5o8o6 


1 


96827 


53o22 1 


88602 


4 


u 


46525 


2 


14940 


2- 


o5485 


5o843 


I 


9 6685 


53o59 1 


88469 


3 


4656o 


2 


14777 


48101 


2- 


o5333 


50879 


I 


96544 


53096 1 
53i34 1 


8833 7 
882o5 


2 


5 9 


465 9 5 


2 


14614 


48737 


2- 


o5i82 


50916 


I 


96402 


1 


6o 


4663 1 


2- 1445 1 


48773 


2 


o5o3o 


50953 


1-96261 


53171 1 


88073 





/ 


Cotang. 


Tangent. 


Cotang. 


T 


ingent. 


Cotang. 


Tangent. 


Cctang. T 


ingent. 


/ 


65° 


64:° 


63° 


62° 



80 NATURAL TANGENTS AND COTANGENTS. 


Tabt.k IIL 


/ 
o 


28° 


29° 


30° 


SP 


' 


Tangent. ( 


Cotang. 


Tangent. C 


'otang. 


Tangent. 


Cotang. 


Tangent. ( 


/Otang. 


53l7I I 


•88073 


5543 1 I 


•8o4o5 


5 77 35 


1^3205 


60086 I 


-66428 


60 


i 


53208 I 


•87941 


55469 I 


80281 


57774 


1-73089 


60126 I 


•663i8 


ll 


a 


53246 I 


.8780; 


55507 I 


80 1 58 


578l3 


L72973 


6oi65 1 


•66209 


3 


53s83 1 


.87677 


55545 I 


8oo34 


57851 


1-72857 


6o2o5 1 


• 66099 


57 


4 


53320 1 


.87546 


55583 1 


7991 1 


57890 


1-7274I 


60245 1 


•65990 
-6588i 


56 


5 


53358 1 


.87415 


55621 1 


79788 


57929 


I -72625 


60284 1 


55 


6 


53395 1 


87283 


55659 1 


79665 


57968 


I -72509 


6o324 1 


•65772 


54 


7 


53432 1 


•87152 


55697 1 


■79542 


58007 


I-72393 


6o364 1 


•65663 


53 


8 


53470 1 


87021 


55736 1 


79419 


58046 


1-72278 


6o4o3 1 


•65554 


52 


9 


535o7 1 


86891 


55774 1 


79296 


58o85 


I-72I63 


6o443 1 


•65445 


5i 


10 


53545 1 


86760 


558i2 1 


79174 


58124 


1-72047 


60483 1 


■65337 


5o 


ii 


53582 1 


8663o 


5585o 1 


79o5i 


58162 


I.7I932 


6o522 I 


■65228 


% 


12 


53620 1 


86499 


55888 1 


78029 


58201 


I.71817 


6o562 1 


65i2o 


i3 


53657 1 


86369 


55926 1 


78807 


58240 


I- 71702 


60602 1 


65on 


47 


14 


53694 1 
53 7 32 1 


8623 9 


55964 1 


78685 


58279 


i-7i588 


60642 1 


64903 


46 


id 


86109 


56oo3 1 


78563 


583 1 8 


I-7I473 


60681 l 


64795 


45 


16 


53769 1 


85979 
8585o 


56o4i 1 


78441 


5835 7 


i-7i358 


60721 1 


64687 


44 


17 


53807 1 


56079 1 


783i 9 


583 9 6 
58435 


1-71244 


60761 1 
60801 1 


64579 


43 


18 


53844 1 


85720 


56117 1 


78198 


1-71129 


64471 


42 


«9 


53882 1 


85591 


56i56 1 


78077 


58474 


i-7ioi5 


60841 1 


64363 


41 


20 


53920 1 


•85462 


56io4 1 


77955 


585i3 


1 -70901 


60881 1 


64256 


40 


21 


53 9 5 7 1 


85333 


56232 1 


77834 


58552 


1-70787 


60921 1 


64148 


is 


22 


5399D 1 


85204 


56270 1 


77713 


585 9 i 


1-70673 


60960 1 


64041 


23 


54o32 1 


85o 7 5 


56309 1 


77592 


5863 1 


i-7o56o 


61000 1 


63g34 


37 


24 


54070 1 


84946 


56347 1 


77471 


58670 


1-70446 


61040 1 


63826 


36 


2D 


54107 1 


84818 


56385 1 


7735i 


58709 


i-7o332 


61080 1 


63719 


35 


26 


54i45 1 


84689 


56424 1 


7723o 


58 7 48 


1. 70219 


61 120 1 


636i2 


34 


2 


54i83 1 


8456i 


56462 1 


77110 


58787 


1.70106 


61160 1 


635o5 


33 


54220 1 


84433 


565oo 1 


76990 
76869 


58826 


1-69992 


61200 1 


633 9 8 


32 


29 


54258 1 


843o5 


5653 9 1 


58865 


1-69879 


61240 1 


63292 


3i 


3o 


54296 1 


84177 


565 77 1 


76749 


58904 


1-69766 


61280 1 


63 1 85 


3o 


3i 


54333 1 


84049 


566i6 1 


7663o 


58 9 44 


1 -69653 


6i32o 1 


63079 


3 


32 


54371 1 


83922 


56654 1 


765io 


58 9 83 


1-69541 


6i36o 1 


62972 


33 


54409 1 


83 79 4 


566 9 3 1 


76390 


59022 


1-69428 


61400 1 


62866 


27 


34 


54446 1 


83667 


56 7 3i 1 


76271 


59061 


1-69316 


61440 1 


62760 


26 


35 


54484 1 


83540 


56769 1 


76131 


59101 


1-69203 


61480 1 


62654 


25 


36 


54522 1 


834i3 


568o8 1 


76032 


59140 


1 -69091 


6i52o 1 


62548' 


24 


37 


5456o 1 


83286 


56846 1 


75913 


59179 


1-68979 


6i56i 1 


62442 


23 


38 


54597 1 
54635 1 


83 1 59 
83o33 


56885 1 


7 5 794 


59218 


1-68866 


61601 1 


62336 


22 


3 9 


56923 1 


756 7 5 


59258 


1-68754 


61641 1 


62230 


21 


4o 


54673 1 


82906 


56962 1 


75556 


59297 
59336 


1-68643 


61681 1 


6.2125 


20 


4i 


54i 1 1 1 


82780 


57000 1 


75437 


1-68531 


61721 1 


62019 


3 


42 


54748 1 


82654 


57039 1 


75319 


59376 


1-68419 


61761 1 


61914 
61808 


43 


54786 1 


82528 


57078 1 


75200 


594i5 


i-683o8 


61801 1 


17 


44 


54824 1 


82402 


57 11 6 1 


75082 


59454 


1-68196 


61842 1 


61703 


16 


45 


54862 1 


82276 


57i55 1 


74964 


59494 


i- 68o85 


61882 1 


61598 


i5 


46 


54900 1 


82i5o 


57193 1 


74846 


5 9 533 


1-67974 


61922 1 


61493 


14 


% 


54938 1 


82025 


57232 1 


74728 


59573 


1.67S63 


61962 1 


6i388 


i3 


D4975 1 


81899 


57271 1 


74610 


59612 


1-67752 


62oo3 1 


6i283 


12 


49 


55oi3 1 


81774 


57309 1 


74492 


59651 


1-67641 


62043 1 • 


61179 


1 1 


5o 


55o5i 1 


81649 


5 7 348 1 


74375 


59691 
59730 


i-6753o 


62o83 1- 


61074 


10 


5i 


55o8 9 1 


8i524 


57386 1 


74257 


1-67419 


62124 i- 


60970 


I 


52 


55i 27 1 


81399 


57425 1 


74i4o 


59770 


1-67309 


62164 i- 


6o865 


53 


55i65 1 


81274 


57464 1 


74022 


59809 


1-67198 


62204 i- 


60761 


1 


54 


552o3 1 


8u5o 


575o3 1 


73905 


59849 


1-67088 


62245 i- 


60657 


6 


55 


5524i 1 


8io25 


57541 1 


73788 


59888 


1-66978 


62285 i- 


6o553 


5 


56 


55279 l 


80901 


5758o 1 ■ 


73671 


59928 


1-66867 


62325 i- 


60449 


4 


5 1 
58 


55317 * 


80777 


57619 i- 


73555 


59967 


1-66757 


62366 1 - 


6o343 


3 


55355 1 


8o653 


57657 1 • 


7 3438 


60007 


1.66647 


62406 1 • 


60241 


2 


5 9 


553 9 3 1 
55431 I- 


80529 


57696 1 • 
5 77 35 1. 


73321 


60046 


1-66538 


62446 1 • 


60137 


1 


60 


8040J 


732o5 


60086 


1-66428 


624S7 1 • 


6co33 





t 


Cotang, T 


ingent. 


Cotang. T 


ingent. 


Cotang. 


Tangent. 


Cotang t Tj 


ingent. 


/ 


61° 


60° 


5 


}° 


58° 



Table III. 


NATURAL TANGENTS AND COTANGENTS. 81 


r 


32° 


33° 


34° 


35° 


t 


Tangent. C 


otang. 


Tangent. C 


'otang. 


Tangent. 


Cotang. 


Tar gent. C 


otang. 


o 


62487 1 


•6oo33 


64941 I 


53 9 86 


6745l 


1-48256 


7C02I I 


428 1 5 


60 


I 


62527 I 


•59930 
.59826 


64982 I 


53888 


67493 


1 .48163 


70064 1 


42726 


% 


2 


62568 I 


65o23 1 


53791 


67536 


I .48070 


70107 1 


42638 


3 


62608 I 


59723 


65o65 1 


536 9 3 


67578 


I-47Q77 

1-47885 


70i5i 1 


4255o 


5 7 


4 


62649 * 


59620 


65 1 06 1 


.53595 


67620 


70194 1 


42462 


56 


5 


62689 I 


59D17 


65i48 1 


.53497 ', 67663 


1-47792 


70238 1 


42374 
42286 


55 


6 


62730 1 


59414 


65i8 9 1 


• 53400 


67702 


1 .47699 


70281 1 


54 


7 


62770 I 


593 1 1 


6523i 1 


•53302 


67748 


1-47607 


7 o325 1 


42198 


53 


8 


6281 1 I 


59208 


65272 1 


•53205 


67790 
6 7 832 


1-47514 


70368 1 


42110 


52 


9 


62852 1 


5910D 


653i4 1 


.53107 


1-47422 


70412 1 


42022 


5i 


IO 


62892 1 
62933 1 


5ooo2 


65355 1 


•53oio 


67875 


1 «4733o 


70455 1 


41934 

41847 


5o 


1 1 


58900 


653 97 * 
65438 1 


52 9 i3 


67917 


1-47238 


70499 » 


40 


12 


62973 1 


58 797 


.52816 


67960 


1. 47U6 


70542 1 


4n59 


48 


i3 


63oi4 1 


586 9 5 


6548o 1 


52719 


68002 


i.47o53 


70586 1 


41672 
4 1 584 


47 


14 


63o55 1 


585 9 3 


65521 1 


.52622 


68o45 


1-46962 
1-46870 


70629 1 
7067J 1 


46 


i5 


63095 1 


58490 


65563 1 


52525 


68088 


4i497 


45 


16 


63i36 1 


58388 


656o4 1 


52429 


68i3o 


1.46778 


70717 1 


41409 


44 


17 


63177 * 


58286 


65646 1 


52332 


68173 


1.46686 


70760 1 


4l322 


43 


18 


63217 1 


58 1 84 


65688 1 


•52235 


68215 


1-46595 


70804 1 


41235 


42 


J 9 


63258 1 


58o83 


65729 1 


52139 
52043 


68258 


i-465o3 


70848 1 


41148 


41 


20 


63299 1 


5 7 o8i 


65771 1 


683oi 


r -46411 


70891 1 


41061 


40 


21 


63340 1 


57879 
57778 


658i3 1 


5 1 946 


68343 


1-46320 


70935 1 


40974 


ll 


22 


6338o 1 


65854 1 


5i85o 


68386 


1-46229 


70979 1 
71023 1 


40887 


23 


63421 1 


57676 


658 9 6 1 
65 9 38 1 


51754 


68429 


1 -46137 


40800 


37 


24 


63462 1 


57575 


5i658 


68471 


1-46046 


71066 1 


40714 


36 


25 


635o3 1 


57474 


65 9 8o 1 


5 1 562 


685i4 


i-45o55 


71110 1 


40627 


35 


26 


63544 1 


57372 


66021 1 


5 1 466 


98557 


1-45864 


7ii54 1 


4o54o 


34 


3 


63584 1 


57271 


66o63 1 


5i37o 


68600 


1-45773 


71198 1 


4o454 


33 


63625 1 


57170 


66io5 1 


51275 


68642 


1-45682 


71242 1 


40367 


32 


29 


63666 1 


57069 


66147 1 


51179 
5 1084 


68685 


1-45592 


71285 1 


40281 


3i 


3o 


63707 1 


56969 


66189 * 


68728 


i-455oi 


7i32 9 1 


40195 


3o 


3i 


63748 1 


56868 


6623o 1 


5oo88 
50893 


68771 


i-454io 


7i3 7 3 1 


40109 


3 


32 


63789 1 


56767 


66272 1 


68814 


1-45320 


71417 1 


40022 


33 


6383o 1 


5666 7 


663 14 1 


50797 


. 6885 7 


1-45229 


71461 1 


3 9 o36 


27 


34 


63871 1 


56566 


66356 1 


50702 


68900 


i-45i39 


7i5o5 1 


39860 


26 


35 


63912 1 


56466 


663 9 8 1 


50607 


68942 


1 • 45o49 


7i549 1 
7 1 5&3 1 
71637 1 


39764 


25 


36 


63953 1 


56366 


66440 1 


5o5i2 


68 9 85 


i-44q58 


39679 


24 


u 


63994 1 
64o35 1 


56265 


66482 1 


5o4n 


69028 


1-44868 


39693 


23 


56i65 


66524 1 


5o322 


69071 


1-44778 


71681 1 


39507 


22 


39 


64076 1 


56o65 


66566 1 


50228 


69114 


1-44688 


71725 1 


39421 


21 


4o 


641 17 1 


55966 
55866 


66608 1 


5oi33 


69157 


1-44598 


71769 1 
7181J 1 


3 9 336 


20 


4i 


641 58 1 


6665o 1 


5oo38 


69200 


1 -44608 


39250 


19 


42 


64199 1 


55766 


66692 1 
667J4 1 


49944 


69243 


i-444i8 


7 i85 7 1 


39165 


18 


43 


64240 1 


55666 


49849 
49755 


69286 


1-44329 


71901 1 


39079 


\l 


44 


64281 1 


5556 7 


66776 1 
66818 1 


69329 


1-44239 


71946 1 


38994 


45 


64322 1 


55467 


49661 


69372 


I-44U9 


71990 1 


38909 


i5 


46 


64363 1 


55368 


66860 1 


49566 


69416 


1 -44o6o 


72o34 1 


38824 


14 


48 

Is 


64404 1 


55269 


66902 1 


49472 


69459 


1-43970 
1-4388 1 


72078 1 


38738 


i3 


64446 1 


55170 


66944 1 


49378 


69502 


72122 1 


38653 


12 


64487 1 ■ 
64528 1 


55071 


66986 1 


49284 


69545 


1-43792 


72166 1 


38568 


1 1 


54072 
54873 


67028 1 


49190 


69588 


1 -43703 


72211 1 


38484 


10 


5i 


64569 1 


67071 1 


49097 


69631 


1-43614 


72255 1 


383 99 


I 


52 


64610 1 


54774 


67113 1 


4qoo3 


69675 


1-43525 


72299 1 


383 14 


53 


64652 1 


54675 


6 7 i55 1 


48909 


69718 


1-43436 


72344 1 


38229 


I 


54 


64693 1 
64734 1 


54576 


67197 1 


48816 


69761 


1-43347 


72388 1 


38i45 


55 


54478 


67239 1 


48722 


69804 


1-43258 


72432 1 


38o6o 


5 


56 


64776 1 


54379 


67282 1 


48629 


69847 


1-43169 


72477 1 


37976 


4 


a 


64817 1 


54281 


67324 1 


48536 


69891 


i-43o8o 


72021 1 


37891 


3 


64858 1 


54i53 


67366 1 


48442 


69934 


1-42992 


72565 1 


37807 


2 


5 9 


64899 1 


54o85 


67409 1 


48349 


69977 


1-42903 


72610 1 


37722 


1 


60 


64941 1 


53986 


67451 1 


48256 


70021 


1.4281b 


72654 1 


37638 





/ 


Cotang. T 


angent. 


Cotang. T 


angent. 


Cciang. 


Tangent. 


Cotang. T 


ingent. 




57° 




56° 


5 


5° 


54° 



42 



32 NATURAL TANGENTS AXD COTANGENTS^ 


Table TIL 


/ 


36° 




37° 


38° 


39° 


' 


Tangent. C 


'otang. 


Tangent. C 


otang. 


1 

Tangent. C 


'otang. 


Tangent. 


Cotang. 





72654 I 


3 7 638 


75355 1 


32704 


78129 1 I 


27994 


80Q-8 


I-23490 


60 


i 


72699 1 


37554 


70401 I 


32624 


78173 I 


27917 


81027 


J - 23 4i6 


Is 


2 


72743 1 


37470 


75447 I 


32544 


78222 I 


27841 


8107O 


1-23343 


3 


72788 I 


3 7 386 


75492 I 
75538 1 


32464 


7826c I 


27764 


1 8l 123 


I -23270 


57 


4 


72832 I 


37302 


32384 


783x6 1 


27688 


8117) 


1 -23iq6 


56 


5 


72877 I 


37218 


75584 1 


323o4 


7 8363 1 


27611 


81220 


1 -23l23 


55 


6 


72921 1 


3 7 i34 


75629 1 


32224 


78410 1 


27535 


81268 


1 -23o5o 


54 


I 


72966 I 


37050 


75670 1 


32144 


78457 1 


27458 


8i3i6 


1-22977 


53 


73oio I 


36961 


75721 1 


32064 


78504 1 


27382 


8 1 364 


I -22G04 


52 


9 


73o55 1 


36883 


75767 1 


31984 


■)855i 1 


27306 


8i4i3 


I-2283I 


01 


10 


73ioo 1 


368oo 


75812 1 


31904 


78598 1 


27230 


81461 


1 -227581 5o 


ii 


73 i44 1 


36716 


75858 1 


3i825 


78645 1 


27i53 


8i5io 


1-22685 I 49 
1-22612 48 


12 


73189 1 


36633 


75904 1 


3i745 


78692 1 
78739 1 


27077 


81008 


i3 


73234 1 


36549 


75950 1 


3i666 


27001 


81606 


1-22539 47 


14 


73278 1 


36466 


75996 1 


3 1 086 


78786 1 


26925 
26849 


8i65o 


I -22467 


46 


ID 


73323 1 


36383 


76042 1 


3i5o7 


78834 1 


81703 


1-22394 


45 


16 


73368 1 


363oo 


76088 1 


3i457 


78881 1 


26774 


8i 7 52 


I-2232I 1 44 


17 


734i3 1 


36217 


76134 1 


3 1 348 


78928 1 


26698 


81800 


I -22249 


43 


18 


73457 1 


36i33 


76180 1 


31269 


78975 1 


26622 


81849 
81898 


I -22176 


42 


19 


735o2 1 


36o5i 


76226 1 


3 1 190 


79022 1 


26546 


I -22104 


41 


20 


73547 1 


35 9 68 
35885 


76272 1 


3mo 


79070 1 


26471 


81946 


I -22o3l 


40 


21 


73592 1 


763i8 1 


3io3i 


79U7 1 


26395 


81995 


1-21 959 


ll 


22 


7363 7 1 


358o2 


76364 1 


30952 


79164 1 


26319 


82044 


1-21886 


23 


7368i 1 


35719 


76410 1 


30873 


79212 1 


26244 


82092 


1-21814 


11 


24 


73726 1 


35637 


76406 1 


30795 


79209 1 
79306 1 


26169 
26093 


82141 


1 -21742 


25 


73771 1 


35554 


76502 1 


30716 


82100 


1-21670 


35 


26 


738i6 1 


35472 


76048 1 


3o637 
3o558 


79354 1 


26018 


82238 


1-21598 


34 


11 


7386i 1 


35389 


76594 1 


794oi 1 


25o43 


82287 


1 • 2ID26 


33 


73906 1 


353o7 


76640 1 


3o48o 


79449 1 


25867 


82336 


1-21454 


3a 


29 


73g5i 1 


35224 


76686 1 


3o4oi 


79496 1 


25792 


82385 


I-2I382 


3i 


3o 


73996 1 


35i42 


76733 1 


3o323 


79544 1 


25717 


82434 


I -2l3lO 


3o 


3i 


74041 1 


35o6o 


76779 1 


3o244 


7 9 5 9 i 1 
7 9 63 9 1 


25642 


824?3 


I-2I238 


3 


32 


74086 1 


34078 


76825 1 


3oi66 


25567 


8203i 


1-21166 


33 


74i3i 1 


34896 


76871 1 


300S7 


79686 1 


20492 


82080 


I • 2 1 094 


27 


34 


74176 1 


348U 


76918 1 


30009 


79734 1 


25417 


82629 


I -21023 


26 


35 


74221 1 


34732 


76964 1 


29931 


79781 1 


25343 


82678 


1 -20951 
1 -20879 
I .20808 


25 


36 


74267 1 


3465o 


77010 1 


29853 


79829 1 


25268 


82727 


24 


3? 


743 1 2 1 


34568 


77057 1 


29775 


79877 1 


25193 


82776 


23 


38 


74357 1 


34487 


77103 1 


29696 


79924 1 


25u8 


82S25 


1 -20736 


22 


3 9 


74402 1 


34405 


77149 1 


29618 


79972 1 
80020 1 


25o44 


82S-4 


1 • 2o665 


21 


4o 


74447 1 


34323 


77196 .1 


29541 


24969 


82923 


1 -2c5g3 


20 


4i 


74402 1 
74538 1 


34242 


77242 1 


29463 ! 


80067 1 


24895 


82972 


I -20522 


a 


42 


34160 


772S9 1 


29385 i 


801 i5 1 


24S20 


83022 


I -2045 1 


43 


74583 1 


34079 
33998 


7733o 1 


29307 : 


801 63 1 


24746 


83071 


1 • 20379 


17 


44 


74628 1 


77 382 1 


29229 


80211 1 


24672 


83 1 20 


1 ■ 2o3o8 


16 


45 


74674 1 


33916 


77428 1 


29152 


80258 1 


24597 


83i6g 


1-20237 


i5 


46 


74719 » 


33835 


77475 1 


29074 


8o3o6 1 


24023 


832i8 


I -20166 


14 


47 


74764 1 


33 7 54 


77021 1 


2S997 ; 


8o354 1 


24449 


83268 


I • 20095 


i3 


48 


74810 1 


33673 


77568 1 


28919 ' 
28842 


80402 1 


24370 


833n 


I -20024 


12 


49 


74855 1 


33592 


77615 1 


8o45c 1 


243oi 


83366 


1-19953 


1 1 


5o 


74900 1 


335u 


77661 1 


28764 


80498 1 


24227 


S34i5 




10 


5i 


74946 1 


3343o 


777 c8 1 


2S6S7 


8o546 1 


24i53 


83465 


1-19811 


9 


52 


74991 1 
70037 1 


33349 


77754 1 


2S610 


S0594 1 


24079 


835i4 


I-IQ-40 


8 


53 


33268 


77801 1 


2S533 


80642 1 


24000 


83564 


1 • 19669 


7 


54 


75082 1 


33i8 7 


77848 1 


28456 


80690 1 
8073S 1 


23o3i 


836 1 3 


1-19599 6 


55 


75128 1 


33107 


77895 1 


28379 


23858 


83662 


1-190281 D 


56 


75173 1 


33o26 


77941 1 


28302 


80786 1 


a3 7 84 


83 7 i2 


I-I9457I 4 


n 


75219 1 


32946 


77988 1 


2S225 


8o834 1 


23710 


83 7 6i 


1.193S7! 3 


75264 1 


32S65 


78o35 1 


2S14S 


80882 1 


23637 
23563 


80S11 


1 - 1 93 1 6 1 2 


5 9 


753io 1 


32785 


780S2 1 


28071 


80930 1 


83S6o 


1 • 19246 j 1 


6o 


75355 1 


32704 


78129 ! 1 


27994 : 


80978 1 


23490 j 80910 


1-191751 


/ 


Cotang. T 


angent. 


Cotang. , T 


angent. 


Cotang. T 


angent. 


Cotang. 


Tangent. 


t 




53° 


52° 


J 


51 C 




5 


0° 





Table III. 


NATURAL TANGENTS AND COTANGENTS. 83 


t 


40° 


41° 


42° 


43° 


60 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. Cotang. 


o 


83910 


I-19I75 


86929 


i-i5o37 


90040 


i-iio6i 


93252 I 


07237 


i 


83 9 6o 


1 .19105 


86980 


1-14969 


90093 


1-10996 
i-ioo3i 


g33o6 1 


07174 


5q 


2 


84009 


I -19035 


8io3i 


1 • 14902 


90146 


g336o 1 


071 12 


58 


3 


84059 
84108 


1-18964 


87082 


1 - 14834 


90199 
90251 


1-10867 


934i5 1 


07049 


&7 


4 


I-l88 9 4 


87133 


1-14767 


1 -10802 


93469 1 


06987 


56 


5 


84i58 


I- 1 8824 


87184 


1 -14699 


9o3o4 


1. 10737 


93624 1 


06923 


55 


6 


84208 


I- I 8 7 54 


87236 


i- 14632 


90357 


1-10672 


93578 1 


06862 


54 


7 


84258 


1-18684 


87287 

8 7 338 


i- 14565 


90410 


1-10607 


93633 1 


06800 


53 


8 


84307 


l-l86l4 


1-14498 


90463 


i-io543 


9 3688 1 


06738 


5a 


9 


8435 7 


I- 18544 


8 7 38 9 


i-i443o 


90616 


1-10478 


93742 1 


06676 


5i 


IO 


84407 


I-I8474 


87441 


i- 1 4363 


9o56g 


1-10414 


93797 1 
9 3852 1 


066 1 3 


5o 


ii 


8445 7 


I.l8404 


87492 


1-14296 


90621 


1 • 1 0349 


o655i 


% 


12 


84507 


i-i8334 


87543 


1-14229 


90674 


1. 10285 


93906 1 


06489 


i3 


84556 


1-18264 


87595 


1-14162 


90727 


I-I0220 


93961 1 


06427 
o6365 


% 


i4 


84606 


1-18194 


87646 


1 -14095 


90781 


i-ioi56 


94016 1 


i5 


84656 


i- 18125 


87698 


1-14028 


90834 


1-10091 


94071 1 


o63o3 


45 


16 


84706 


i-i8o55 


87749 


1-13961 


90887 


1-10027 


94125 1 


06241 


44 


17 


84756 


1-17986 


87801 


i-i38 9 4 


90940 


1-09963 


94180 1 


06179 


43 


18 


84806 


1.17916 


87852 


i- i38 2 8 


90993 


1-09899 


94235 1 


06117 


42 


19 


84856 


1. 1 7846 


87904 


1-13761 


91046 


1-09834 


94290 1 


o6o56 


41 


20 


84906 


1. 17777 
1-17708 


87955 


1 -13694 


91099 


1-09770 


94345 1 


05994 


40 


21 


84g56 


88007 


I-I3627 


91 153 


1-09706 


94400 1 


05932 


ll 


22 


85oo6 


i- 17638 


88o5 9 


i-i356i 


91206 


1-09642 


94455 1 


05870 


23 


85o57 


1-17569 


88110 


i- 13494 


91259 


I -09578 


945io 1 


05809 


37 


24 


85io 7 


1 -i75oo 


88162 


i-i3428 


9i3i3 


1-09514 


94565 1 


05747 


36 


25 


85i5 7 


1 -i743o 


88214 


i-i336i 


9 1 366 


1-09450 


94620 1 


o5685 


35 


26 


852o 7 


1-17361 


88265 


1-13295 


91419 
9U73 


1.09386 


94676 1 


o5624 


34 


s 


8525 7 


1-17292 


883 1 7 


I-I3228 


1-09322 


9473i 1 


o5562 


33 


853o 7 
85358 


1-17223 


8836 9 


1 • i3i62 


91526 


1-09258 


94786 1 


o55oi 


32 


2 9 


1-17154 


88421 


1 • 13096 


9i58o 


I -09195 


94841 1 


05439 


3i 


3o 


854o8 


1-17085 


88473 


1 • 13029 


9i633 


1 -09i3i 


94896 1 


05378 


3o 


3i 


85458 


1-17016 


88524 


i- 12963 


91687 


1 -09067 


94952 1 


o53i7 


3 


32 


85509 


i- 16047 
1. 16878 


885 7 6 


1-12897 
i-i283i 


91740 


1 -09003 


95007 1 


o5255 


33 


8555 9 


88628 


91794 


1-08940 


95062 1 


05194 


27 


34 


856o 9 


1-16809 


88680 


1-12765 


91847 


1-08876 


9 5n8 1 


o5i33 


26 


35 


8566o 


1-16741 


88732 


1-12699 
1 -12633 


91901 


i- 0881 3 


95173 1 


05072 


25 


36 


85710 


1 • 16672 


88784 


91955 


1-08749 
1-08686 


95229 1 


oSoio 


24 


32 


85761 


i-i66o3 


88836 


1 -12567 


92008 


95284 1 


04949 


23 


858n 


1. 16535 


88888 


I-I250I 


92062 


1-08622 


95340 1 


04888 


22 


3 9 


85862 


1-16466 


88940 


I- 1 2435 


92116 


i-o855 9 


953o5 1 


04827 


21 


4o 


85912 


1 - 163 9 8 


88992 


I-I236Q 


92170 


1-08496 


g545i 1 


04766 


20 


4i 


85 9 63 


1 -16329 


89045 


i-i23o3 


92223 


1-08432 


955o6 1 


04705 


;g 


42 


86014 


1-16261 


89097 


1- 12238 


92277 


i-o836 9 


95562 1 


04644 


43 


86064 


1 -16192 


89149 


1-12172 


9233i 


i-o83o6 


9 56i8 1 


04583 


17 


44 


861 1 5 


1 • 16124 


89201 


I • ! 2 1 06 


92385 


1-08243 


95673 1 


04522 


16 


45 


86166 


1 -i6o56 


89253 


1-12041 


92439 


1-08179 


95729 1 


04461 


i5 


46 


86216 


1. i 5987 


8 9 3o6 


I • 1 1975 


92493 


1-08116 


95785 1 


04401 


14 


47 


86267 


1-15919 


8 9 358 


1-11909 

1.11844 


92547 


i-o8o53 


95841 1 


04340 


i3 


48 


863 1 8 


i-i585i 


89410 


926or 


1-07990 


95897 1 


04279 
04218 


12 


P 


86368 


I-I5783 


8 9 463 


I-1I778 


92655 


1-07927 


95952 1 


1 1 


5o 


86419 


I«i57i5 


8 9 5 1 5 


1 -11713 


92709 


1-07864 


96008 1 


041 58 


10 


5i 


86470 


i- 1 5647 


89567 


1-11648 


92763 


1 -07801 


96064 1 


04097 


i 


52 


86521 


I- 15579 


89620 


i-n582 


92817 


1-07738 


96120 1 


o4o36 


53 


865 7 2 


i-i55ii 


89672 


i-n5i7 


92872 


1 -07676 


96176 1 


03976 


7 


54 


86623 


i • 1 5443 


89723 


I-H452 


92926 


1 -07613 


96232 1 


o3oi5 
o3855 


6 


55 


86674 


1 • 15370 


89777 


1. 11387 


92980 


1 -07550 


96288 1 


5 


56 


86725 


i-i53o8 


8 9 83o 


I-II32I 


93o34 


1-07487 


96344 1 


03794 


4 


& 


86776 


i-i524o 


8 9 883 


1 • 1 1256 


9 3o88 


1 -07425 


96400 1 


03734 


3 


86827 


i-i5i72 


89935 


1-11191 


93i43 


1 -07362 


96457 1 


03674 


2 


5 9 


86878 


i-i5io4 


89988 


1 • i 1 1 26 


93197 
93252 


1-07299 


965i3 1 


o36i3 


1 


6o 


86929 


i-i5o37 


90040 


1 • 1 1 06 1 


1-07237 


96569 1 


o3553 




/ 


( otang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. 


Tangent. 


Cotang. T 


angent. 




4 


9 U 


4 


8° 


4 


70 


46° 



84 



NATURAL TANGENTS AND COTANGENTS. 



Table III. 



44° 



2 
3 
4 
5 
6 

7 
8 

9 

10 

ii 

12 

i3 
14 
i5 

16 
17 
18 

19 

20 
21 
22 
23 
24 
25 
26 

27 
28 

29 
3o 



96569 
96625 

96681 
96738 
96794 
96800 
96907 

96963 
97020 

97076 

97133 

97189 

97246 
97302 

97359 
97416 

97472 
97329 

97586 
97643 
97700 
97756 
978.3 
97870 

97927 
97984 
98041 
98098 
9 8i55 
9 82i3 
98270 



Cotang. 



Cotang. 



i-o3553 
i'o3493 
1-03433 
1-03372 
i-o33ia 

1-03252 

I- 03192 
i-o3i32 
i- 03072 

I-030I2 

I -02052 
1-02892 
1-02832 

1-02772 
1-02713 
1-02653 

1 -02593 
1-02533 
1-02474 
1-02414 
1-02355 
1-02295 

1-02236 

1-02176 
1-02117 
1 • 02057 
1 -01998 
1 01939 
1-01879 
1-01820 
1-01761 



Tangent. 



45 c 



60 
5 9 

58 

57 
56 
55 
54 
53 

52 

5i 

DO 

% 

47 
46 
45 

44 
43 

42 
4i 
4o 
3 9 
38 

37 

36 
35 
34 
33 

32 

3i 

3o 



47 

48 

49 
5o 
5i 

52 

53 

54 
55 

56 

57 
58 

5 9 
60 



44< 



Tangent. 



98327 

98384 
98441 

98499 
98556 
9 86i3 
98671 
98728 



98901 
98953 
99016 
99073 
99i3i 

■ 99189 
99247 
99304 
99362 
99420 
99478 
99536 
9 9 5 9 4 
99602 
99710 
99768 
99826 
99884 
99942 
Unit. 



Cotang. 



Cotam 



1-01702 

I-0l642 

i-oi583 

oi524 

oi465 

01406 

01 347 

01288 

01229 

01170 

01112 

oio53 

1 • 00994 

1-00935 

1-00876 

1 -00818 
1-00759 
1-00701 
1-00642 
1 -oo583 
i-oo525 
1 • 00467 
1 -00408 
i-oo35o 
1-00291 
1-00233 
1-00175 

I-OOIIO 

1 -ooo58 

Unit. 



Tangent. 



45° 



TABLE OF CONSTANTS. 

Base of Napier's system of logarithms = s = 2-71 82S 1 82S459 

Mod. of common syst. of logarithms = .... com. log. t = M = 0-434294481903 

Eatio of circumference to diameter of a circle = ....it = 3 • 141 592633590 

log. v = 0-497149872694 

ir = 9-869604401089 y/ rc= I-772453850906 

Arc of same length as radius == 180 -r- n = 10800' •?- * = 648000" -r- n 

180 -T- ir = 57°- 2957795130, log. = 1-758122632409 

10800' -&.-jr= 3437' -7467707849, log. = 3-536273882793 

648000" -r- jr= 206264". 8062470964, log. = 5-3i4423i33i76 

Tropical year = 365d. 5h. 48m. 47s. -588 = 365d. -242217456, log. = 2-56258io 
Sidereal year = 365d. 6h. 9m. 10s. -742 = 365d. -25637433a, log. = 2-5625978 
24h. sol. t.=24h. 3m. 56s. • 555335 sid. t.=24h.Xi .00273791, log. 1.002=0-0011874 
24h.sid.t.=24h.—(3m.55s. .90944) sol. t.=24h.X 0-9972696, log. 0-997=9-9988126 

British imperial gallon = 277-274 cubic inches, log. = 2-4429091 

Length of sec. pend., in inches, at London, 39-13929; Paris, 39-1285; New 

York, 39-1285. 
French metre = 3-2808992 English/^ = 39-3707904 inches. 
1 cubic inch of water (bar. 3o inches, Fahr. therm. 62 ) = 232-458 Troy grains. 





A TABLE OF MEAN REFRACTIONS IN DECLINATION. 


85 


w 
J 

2 

« 
O 


Refraction in Declination. 


For Latitude 15 . 


+ 30° 


+ 15° 


+ 10° 


+ 5° 


0° 


— 5° 


— 10° 


— 15° 


— 80° 

40" 


oh. 


-05" 


0" 


+ 05" 


10" 


15" 


21" 


27" 


33" 


2 


—03 


+ 02 


07 


12 


18 


23 


29 


36 


43 


3 


+ 01 


05 


11 


16 


22 


28 


34 


4 1 


49 


4 


08 


12 


19 


24 


30 


37 


44 


53 


1 04 


5 


29 


34 


4 1 


49 


59 


1 10 


1 24 


143 


2 '08 


For Latitude 17 30'. 


oh. 


— 02" 


+ 02" 


08" 


13" 


18" 


24 


3° 


36" 


44" 


2 





05 


10 


15 


21 


27 


33 


40 


48 


3 


+ 02 


10 


15 


21 


27 


33 


40 


48 


57 


4 


13 


18 


23 


29 


35 


43 


5i 


i'oi 


1 13 


S 


34 


4i 


49 


58 


1 10 


1 23 


i' 4 i 


2 06 


2 42 


For Latitude 20 . 


oh. 


0" 


05' 


10" 


X5" 


21" 


27" 


33 


4° 


48" 


2 


03 


07 


13 


18 


24 


30 


36 


44 


52 


3 


06 


13 


18 


24 


30 


36 


44 


52 


l'o2 


4 


17 


22 


28 


35 


42 


50 


i'oo 


in 


I 26 


5 


39 


47 


57 


107 


1 '20 


1 37 


2 00 


2 32 


3 25 








I 


"or Latitude 22 30'. 








oh. 


02" 


08" 


13" 


18" 


24' 


3° 


36" 


44" 


52" 


2 


06 


11 


15 


21 


27 


33 


40 


48 


57 


3 


11 


15 


21 


27 


33 


40 


48 


57 


i'o8 


4 


20 


26 


32 


39 


46 


56 


1 '07 


1 19 


1 37 


5 


45 


53 


1 03 


i'i6 


1 3i 


l'52. 


2 21 


307 


4 28 


For Latitude 25 . 


oh. 


05" 


10" 


15" 


21" 


27" 


33" 


40 


48" 


,57" 


2 


08 


14 


19 


25 


3 1 


38 


46 


54 


1 05 


3 


12 


18 


24 


30 


37 


44 


53 


1 '04 


1 18 


4 


23 


29 


35 


45 


53 


1 03 


i'i6 


1 3i 


1 52 


5 


49 


59 


i'io 


l'24 


r'52 


2 07 


2 44 


3 46 


5 43 


For Latitude 27 30'. 


oh. 


08" 


13" 


18" 


24" 


3° 


36" 


44" 


52" 


1 '02" 


2 


11 


16 


22 


28 


34 


4i 


49 


i'oo 


1 10 


3 


17 


22 


28 


35 


42 


50 


i'oo 


I II 


1 26 


4 


28 


35 


42 


50 


1 '00 


in 


I 26 


1 43 


2 09 


5 


54 


1 05 


i'i8 


1 34 


1 54 


2 24 


3 11 


4 38 


815 


For Latitude 30 . 


oh. 


10" 


15" 


21" 


27" 


33" 


40" 


48" 


, 57 


i'o8" 


2 


14 


19 


25 


3i 


38 


46 


54 


105 


1 18 


3 


20 


26 


32 


39 


47 


55 


106 


1 19 


136 


4 


32 


39 


46 


52 


i'o6 


i'i 9 


1 35 


157 


2 29 


5 


1 00 


i'io 


l'24 


i'52 


2 07 


2 44 


3 46 


5 43 


13 06 


For Latitude 32 30'. 


oh. 


13 


18" 


24" 


3° 


36" 


44" 


52" 


z'oTt" 


i'i4'' 


2 


17 


22 


28 


35 


42 


50 


1 00 


I II 


1 26 


3 


23 


29 


35 


43 


51 


i'oi 


1 13 


128 


1 47 


4 


35 


43 


5i 


i'oi 


1 13 


I 27 


1 46 


2 13 


2 54 


5 


1 03 


1 15 


I3 1 


1 53 


2 20 


305 


4 25 


7 36 




For Latitude 35 . 


oh. 


15 


21" 


27 


33 


4° 


48" 


57" 


I '08" 


l'2l" 


2 


20 


25 


32 


38 


46 


55 


1 '05 


I l8 


1 35 


3 


26 


33 


39 


47 


56 


i'o7 


1 21 


138 


2 00 


4 


39 


47 


, 56 


107 


1 20 


136 


1 59 


2 32 


325 


5 


1 '07 


1 20 


1 '38 


2 00 


2 34 


3 29 


5 14 


IO l6 





86 






REFRACTION IN DECLINATION. 






W<3 


For Latitude 37 30'. 


+ 30° 


+ 15° 


+ 10° 


+ 5° 0° 


— 5° 


— 10° 


— 15° 


— 20° 


oh. 


18" 


24" 


30" 


36" 


44" 


52" 


l'o2." 


i'i 4 // 


x'tzg" 


2 


22 


28 


35 


42 


50 


1 00 


I 12 


1 26 


1 45 


3 


29 


36 


43 


52 


1 '02 


1 14 


I 29 


i49 


2 16 


4 


, 43 


5i 


1 01 


"3 


1 27 


1 49 


2 14 


254 


405 


5 


i'ii 


l'26 


1 54 


2 10 


2 49 


3 55 


6 is 


1458 




For Latitude 40 . 


oh. 


21" 


27" 


33 


40" 


48" I 57" 


i'o8" 


l'2l" 


i'39" 


2 


25 


32 


39 


46 


52 1 1 '06 


1 19 


1 35 


1 57 


3 


33 


40 


48 


57 


i'o8 j 1 21 


138 


2 02 


2 36 


4 


, 47 


55 


1 '06 


1 19 


1 3 6 1 58 


2 30 


321 


4 59 


5 


1 15 


1 31 


1 5i 


2 20 


3 05 ! 4 25 


7 34 


25 18 




For Latitude 42 30'. 


oh. 


24" 


30" 


3°" 


44" 


52" 


x'ca" 


i'i4" 


x'-zg" 


1 '49" 


2 


28 


35 


39 


50 


1 '00 


1 12 


1 26 


145 


2 11 


3 


36 


43 


52 


l'02 


1 13 


1 29 


149 


2 17 


2 59 


4 


50 


1 00 


i'ii 


I 26 


1 44 


2 10 


2 49 


3 55 


6 16 


5 


i'i6 


136 


158 


2 30 


3 22 


5 00 


924 






For Latitude 45 . 


oh. 


27" 


33 


40" 


48" 57" 


i'o8" 


I'2I" 


1 39 


I'cyz" 


2 


32 


39 


46 


52 


1 06 


1 19 


1 35 


1 57 


2 29 . 


3 


4° 


47 


56 


I '07 


1 21 


138 


2 00 


2 34 


329 


4 


54 


1 '04 


i'i6 


1 33 


1 54 


2 24 


3 11 


4 38 


815 


5 


1 23 


141 


2 05 


241 


3 40 


540 


12 02 






For Latitude 47 30'. 


oh. 


3° 


36" 


44" 


52" 


i'o2" i'i4" 


l'2 9 " 


i'49" 


2'l8" 


2 


35 


42 


50 


1 '00 


I 12 


126 


145 


2 01 


2 51 


3 


43 


5 1 


1 '01 


1 13 


I 28 


147 


2 15 


2 56 


408 


4 


56 


109 


1 23 


1 40 


205 


2 40 


3 39 


5 37 


II 18 


5 


l'27 


1 46 


2 12 


2 52 


4 OI 


630 


16 19 






For Latitude 50 . 


oh. 


33" 


40" 


48" 


,57" 


i'o8" 


l'2l" 


1 '39" 


2 '02" 


2'o6" 


2 


38 


46 


55 


1 '06 


1 18 


135 


1 57 


2 28 


3 19 


3 


47 


56 


i'o6 


1 19 


136 


2 29 


231 


3 23 


502 


4 


1 '02 


114 


1 29 


1 48 


2 16 


258 


418 


659 


19 47 


5 


1 30 


1 5i 


2 19 


304 


4 22 


7 28 


24 10 






For Latitude 52 30'. 


oh. 


36" 


44" 


52" 


l'02" 


i'i 4 " I i'zq" 


1 '49" 


2'l8" 


3'°5" 


2 


43 


50 


59 


I II 


1 26 J 1 42 


2 23 


2 49 


3 55 


3 


50 


1 '00 


i'ii 


I 26 


1 45 2 11 


2 51 


258 


6 22 


4 


1 '05 


1 18 


1 35 


2 IO 


2 28 1 3 19 


4 53 


8 42 




5 


1 34 


156 


227 


3 16 


4 47 1 8 52 








For Latitude 55 . 


oh. 


40 


48" 


57" 


i'o8" i'2i" i'39" 


i'oz" 


2'36" 


3 33 


2 


40 


55 


1 '05 


1 18 1 34 


156 


2 30 


3 i5 


4 47 


3 


55 


i'o6 


1 19 


1 35 1 58 


2 30 


321 


4 58 


919 


4 


no 


123 


1 42 


2 06 | 2 43 


3 44 


5 49 


12 41 




5 


1 37 


2 01 


2 34 


3 28 1 5 15 10 iS 








For Latitude 57 30'. 


oh. 


44' 


52" 


iW 


1 14 


I'zg*' 


i'49" 


2'l8" 


3'°5" 


4'37" 


2 


5° 


, 59 


I II 


1 25 


143 


2 09 


2 47 


3 5i 


6 04 


3 


58 


1 10 


I 24 


142 


2 07 


2 43 


3 45 


5 50 


12 47 


4 


i'ii 


1 25 


143 


2 10 


2 50 


3 55 


6 14 


14 49 




5 


I 41 


2 06 


2 42 


342 


5 46 


12 20 








For Latitude 6o°. 


oh. 


48" 


57" 


i'oS" 


i'2i" I i'3g" 


2'02" 


2' 3 6" 


3'33" 




2 


54 


1 '04 


I 17 


1 33 1 54 


2 24 


3 12 


438 


815 


3 


103 


1 15 


1 30 


1 51 2 20 


3 04 


424 


7 3i 




4 


1 18 


1 34 


156 


2 28 3 18 


4 50 


8 53 






5 


145 


2 11 


2 SO 


3 57 6 21 15 32 









TABLES 



FOR OBTAINING- 



HORIZONTAL DISTANCES 



DIFFERENCES OF LEVEL, 



STADIA READINGS. 



88 DISTANCES. 


0° 


1 

oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 


a 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9888 


8.9874 


1.4000 


OI 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9888 


8.9874 


1.4000 


02 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9888 


8.9874 


1.4000 


03 


0.9986 


1.9972 


2.9958 


3-9944 


4-9930 


5.9916 


6.9902 


7.9888 


8.9874 


1.4000 


04 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9888 


8.9874 


1.4000 


05 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9888 


8.9874 


1.4000 


06 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9888 


8.9874 


1.4000 


07 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9888 


8.9873 


1.4000 


08 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9888 


8.9873 


1.4000 


09 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9902 


7.9887 


8.9873 


1.4000 


10 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5.9916 


6.9901 


7.9887 


8.9873 


1.4000 


II 


0.9986 


1.9972 


2.9958 


3-9944 


4.9930 


5-9915 


6.9901 


7.9887 


8.9873 


1.4000 


12 


0.9986 


1.9972 


2.9958 


3-9943 


4.9929 


5-9915 


6.9901 


7.9887 


8.9S73 


1.4000 


13 


0.9986 


1.9972 


2.9958 


3-9943 


4.9929 


5-9915 


6 9901 


7.9887 


8.9873 


1.4000 


14 


0.9986 


1.9972 


2-9957 


3-9943 


4.9929 


5-9915 


6.9901 


7.9887 


8.9872 


1.4000 


15 


0.9986 


1.9972 


2-9957 


3-9943 


4.9929 


5-99 J 5 


6.9901 


7.9886 


8.9S72 


1.4000 


16 


0.9986 


1.9972 


2-9957 


3-9943 


4.9929 


5-9915 


6.9900 


7.9886 


8.9872 


1.4000 


17 


0.9986 


1.9972 


2-9957 


3-9943 


4.9929 


5-99*5 


6.9900 


7.9886 


8.9872 


1.4000 


18 


0.9986 


1.9971 


2-9957 


3-9943 


4.9929 


5-99I4 


6.9900 


7.9886 


8.9872 


1.4000 


19 


0.9986 


1.9971 


2-9957 


3-9943 


4.9929 


5.9914 


6.9900 


7.9886 


8.9871 


1.4000 


20 


0.9986 


1.9971 


2-9957 


3-9943 


4.9928 


5-9914 


6.9900 


7.9885 


8.9871 


1.4000 


21 


0.9986 


1.9971 


2-9957 


3-9943 


4.9928 


5-9914 


6.9899 


7.9885 


8.9871 


1.3999 


22 


0.9986 


1:9971 


2-9957 


3-9942 


4.9928 


5-9913 


6.9899 


7.9885 


8.9870 


1-3999 


23 


0.9986 


1.9971 


2-9957 


3.9942 


4.9928 


5.9913 


6.9899 


7.9884 


8.9870 


1-3999 


24 


0.9985 


1.9971 


29956 


3.9942 


4.9927 


5.9913 


6.9898 


7.9884 


8.9869 


1-3999 


25 


0.9985 


1.9971 


2.9956 


3-9942 


4.99 2 7 


5.9913 


6.9898 


7.9884 


8.9869 


1-3999 


26 


0.9985 


1.9971 


2.9956 


3.9942 


4.9927 


5.9912 


6.9898 


7.9883 


8.9869 


1-3999 


27 


0.9985 


1.9971 


2.9956 


3-9941 


4.9927 


5.9912 


6.9898 


7.9883 


8.9868 


1-3999 


28 


0.9985 


1.9971 


2.9956 


3-9941 


4.9927 


5-99 12 


6.9897 


7.9883 


89868 


1-3999 


29 


0.9985 


1.9971 


2.9956 


3-9941 


4.9926 


5.9912 


6.9897 


7.9882 


8.9868 


1-3999 


30 


c.9985 


1.9970 


2.9956 


3-9941 


4.9926 


5-99" 


6.9897 


7.9882 


8.9867 


1.3999 


31 


0.9985 


1.9970 


2.9956 


3-9941 


4.9926 


5-99II 


6.9896 


7.9881 


8.9S67 


L3999 


32 


0.9985 


1.9970 


2-9955 


3-9940 


4.9026 


5-99" 


6.9896 


7.9S81 


8.9866 


1.3999 


33 


0.9985 


1.9970 


2-9955 


3-9940 


4.9925 


5.9910 


6.9895 


7.9880 


8.9866 


1-3999 


34 


0.9985 


1.9970 


2-9955 


3.9940 


4.9925 


5.9910 


6.9895 


7.9880 


8.9865 


1-3999 


35 


0.9985 


1.9970 


2-9955 


3.9940 


4.9925 


5.9910 


6.9895 


7.9880 


8.9S65 


L3999 


36 


0.9985 


1.9970 


2-9955 


3.9940 


4.9924 


5.9909 


6.9894 


7.9879 


8.9864 


1.3999 


37 


0.9985 


1.9970 


2-9954 


3-9939 


4.9924 


5.9909 


6.9894 


7.9879 


8.9863 


1-3999 


38 


0.9985 


1.9970 


2-9954 


3-9939 


4.9924 


5.9909 


6.9893 


7.9878 


8.9863 


1-3999 


39 


0.9985 


1.9969 


2-9954 


3-9939 


4.9924 


5.990S 


6.9893 


7.9878 


8.9862 


1-3999 


40 


0.9985 


1.9969 


2-9954 


3-9939 


4.9923 


5.990S 


6.9893 


7.9877 


8.9862 


1-3999 


4i 


0.9985 


1.9969 


2-9954 


39938 


4.9923 


5-9907 


6.9892 


7.9877 


8.9861 


1.3998 


42 


0.9984 


1.9969 


2-9953 


3-9938 


4.9922 


5-9907 


6.9S91 


7.9876 


S.9S60 


1.3998 


43 


0.9984 


1.9969 


2-9953 


3-9938 


4.9922 


5-9907 


6.9891 


7.9875 


8.9860 


1.3998 


44 


0.9984 


1.9969 


2-9953 


3-9937 


4.9922 


5.9906 


6.9890 


7.9875 


8.9859 


1.3998 


45 


0.9984 


1.9969 


2-9953 


3-9937 


4.9921 


5.9906 


6.9890 


7.9874 


S.9858 


1.3998 


46 


0.9984 


1.9968 


2-9953 


3-9937 


4.9921 


5-9905 


6.9889 


7-9874 


8.9858 


1.3998 


47 


0.9984 


1.9968 


2.9952 


3-9936 


4.9921 


5-9905 


6.9SS9 


7-9873 


8.9857 


I-399S 


48 


0.9984 


1.9968 


2.9952 


3-9936 


4.9920 


5.9904 


6.98S8 


7.9872 


S.9856 


1.3998 


49 


0.9984 


1.9968 


2.9952 


3-9936 


4.9920 


5.9904 


6.9888 


7.9872 


8.9856 


1.3998 


5o 


0.9984 


1.9968 


2.9952 


3-9936 


4.9919 


5-9903 


6.9887 


7.9871 


S.9S55 


1.3998 


5i 


0.9984 


1.9968 


2.9951 


3-9935 


4.9919 


5-9903 


6.9SS7 


7.9870 


S.9S54 


1-3998 


52 


0.9984 


1.9967 


2.9951 


3 9935 


4.9919 


5.9902 


6.98S6 


7.9870 


S.9S53 


1.3998 


53 


0.9984 


1.9967 


2-995 1 


3-9934 


4.9918 


5.9902 


6.9SS5 


7.9869 


.8.9S52 


1.3998 


54 


0.9984 


1.9967 


2-995 1 


3*9934 


4.9918 


5.9901 


6.9SS5 


7.9S68 


S.9852 


i.yggS 


55 


o.99 s 3 


1.9967 


2.9950 


3-9934 


4.9917 


5.9901 


6.9SS4 


7-9867 


S.9S51 


1.3998 


56 


0.99S3 


1.9967 


2.9950 


3-9933 


4.9917 


5.9900 


6.9SS3 


7.9S67 


S.9S50 


1.399S 


57 


0.9983 


1.9966 


2.9950 


3-9933 


4.9916 


5.9899 


6.9SS3 


7.9S66 


8.9S49 


1.399S 


58 


0.9983 


1.9966 


2.9949 


3-9933 


4.9916 


5.9899 


6.98S2 


7.9S65 


S.9S4S 


1.3998 


59 


0.9983 


1.9966 


2.9949 


3-9932 


4.9915 


5.9S9S 


6.9SS1 


7.9864 


8.9847 


1-3998 


60 


0.9983 


1.9966 


2.9949 


3-9932 


4.9915 


5-9S9S 


6.9SS1 


7.9S64 


S.9S47 


1.3908 



HEIGHTS. 



1 


3 


3 


o.oooo 


0.0000 


0.0000 


0.0003 
0.0006 


0.0006 
0.0012 


0.0009 
0.0017 


0.0009 


0.0017 


0.0026 


0.0012 


0.0023 


0.0035 


0.0015 


0.0029 


0.0044 


0.0017 
0.0020 


0.0035 
0.0041 


0.0052 
0.0061 


0.0023 
0.0026 
0.0029 


0.0046 
0.0052 
0.0058 


0.0070 
0.0078 
0.0087 


0.0032 


0.0064 


0.0096 


0.0035 
0.0038 


0.0070 
0.0076 


0.0105 
0.0113 


0.0041 


0.0081 


0.0122 


0.0044 


0.0087 


0.0131 


0.0046 


0.0093 


0.0139 


0.0049 


0.0099 


0.0148 


0.0052 

0.0055 


0.0105 

O.OIIO 


0.0157 
0.0166 


0.0058 


0.0116 


0.0174 


0.0061 


O.OI22 


0.0183 


0.0064 
0.0067 


0.0128 
0.0134 


0.0192 
0.0200 


0.0070 

0.0073 

0.0076 
0.0078 
0.0081 
0.0084 
0.0087 


0.0139 
0.0145 
0.0151 
0.0157 

0.0163 
0.0168 
0.0174 


0.0209 
0.0218 
0.0227 
0.0235 
0.0244 
0.0253 
0.0261 


0.0090 
0.0093 
0.0096 
0.0099 


0.0180 
0.0186 
0.0192 
0.0198 


0.0270 
0.0279 
0.0288 
0.0296 


0.0102 


0.0203 


0.0305 


0.0105 


0.0209 


0.0314 


0.0107 


0.0215 


0.0322 


O.OIIO 


0.0221 


0.0331 


0.0113 
0.0116 


0.0227 
0.0232 


0.0340 
0.0349 


0.0119 


0.0238 


0.0357 


O.OI22 


0.0244 


0.0366 


0.0125 
0.0128 


0.0250 
0.0256 


0.0375 
0.0383 


0.0131 
0.0134 


0.0261 
0.0267 


0.0392 
0.0401 


0.0137 
0.0139 

0.0142 
0.0145 


0.0273 
0.0279 
0.0285 
0.0290 


0.0410 
0.0418 
0.0427 
0.0436 


0.0148 


0.0296 


0.0444 


0.0151 
0.0154 


0.0302 
0.0308 


0.0453 
0.0462 


0.0157 

0.0160 
0.0163 
0.0166 
0.0168 


0.0314 
0.0319 
0.0325 
0.0331 
0.0337 


0.0470 
0.0479 
0.0488 
0.0497 
0.0505 


0.0171 


0.0343 


0.0514 


0.0174 


0.0349 


0.0523 



0.0000 
O.OOI2 
0.0023 
0.0035 

0.0046 
0.0058 
0.0070 
0.0081 

0.0093 

0.0105 
0.0116 

0.0128 

0.0139 
0.0151 
0.0163 
0.0174 
0.0186 
0.0198 
0.0209 
0.0221 
0.0232 

0.0244 
0.0256 
0.0267 
0.0279 
0.0290 
0.0302 

0.0314 
0.0325 
0.0337 
0.0349 

0.0360 

0.0372 
0.0383 
0.0395 

0.0407 
0.0418 
0.0430 
0.0441 

0.0453 
0.0465 

0.0476 
0.0488 
0.0500 
0.051 1 
0.0523 
0.0534 
0.0546 
0.0558 
0.0569 
0.0581 

0.0592 

0.0604 
0.0616 
0.0627 
0.0639 
0.0650 
0.0662 
0.0674 
0.0685 
0.0697 



0.0000 
0.0015 
0.0029 
0.0044 
0.0058 
0.0073 
0.0087 
0.0102 
0.0116 

0.0131 
0.0145 

0.0160 
0.0174 
0.0189 
0.0203 
0.0218 
0.0232 
0.0247 
0.0261 
0.0276 
0.0290 

0.0305 
0.0320 

0.0334 
0.0349 
0.0363 
0.0378 
0.0392 
0.0407 
0.0421 
0.0436 

0.0450 
0.0465 

0.0479 

0.0494 
0.0508 

0.0523 
0.0537 

0.0552 
0.0566 
0.0581 

0.0595 

0.0610 
0.0624 

0.0639 
0.0654 
0.0668 
0.0683 

0.0697 
0.0712 
0.0726 

0.0741 

0.0755 

0.0770 
0.0784 
0.0799 
0.0813 
0.0828 

0.0842 
0.0857 
0.0871 



G 



0.0000 
O.0017 
0.0035 
0.0052 
0.0070 
0.0087 
0.0105 
9.0122 
0.0139 
0.0157 
0.0174 

0.0192 
0.0209 
0.0227 
0.0244 
0.0261 
O.0279 
0.0296 
0.0314 
0.0331 
0.0349 

0.0366 
0.0383 
O.0401 
0.0418 
0.0436 
0-0453 
0.0471 
0.0488 
O.0505 
0.0523 

0.0540 
O.0558 
0.0575 
0.0593 
0.0610 
0.0627 
0.0645 
0.0662 
0.0680 
0.0697 

0.0715 
O.0732 
0.0749 
O.0767 
0.0784 
0.0802 
O.0819 
0.0836 
0.0854 
0.0871 

0.0889 
0.0906 
0.0923 
0.0941 
0.0958 
0.0976 
0.0993 
O.IOII 

o. 1028 
o. 1046 



0.0000 
0.0020 
0.0041 
0.0061 
0.0081 
0.0102 
O.OI22 
0.0142 
0.0163 
0.0183 
0.0203 

0.0224 
0.0244 
0.0264 
0.0285 
0.0305 
0.0325 
0.0346 
0.0366 
0.0386 
0.0407 

0.0427 

0.0447 

0.0468 
0.0488 
0.0508 
0.0529 
0.0549 
0.0569 
0.0590 

0.0610 

0.0630 
0.0651 
0.0671 
0.0691 
0.0712 
0.0732 
0.0752 
0.0773 
0.0793 

0.0813 

0.0834 
0.0854 
0.0874 
0.0895 
0.0915 
0.0935 
0.0956 
0.0976 
0.0996 
0.1017 

0.1037 
0.1057 
0.1077 

0.1098 
O.I 1 18 
0.1 138 

0.1159 
0.1179 

0.1 199 
0.1220 



0.0000 
0.0023 
0.0046 
0.0070 

0-0093 

0.0116 

0.0139 
0.0163 
0.0186 
0.0209 
0.0232 

0.0256 
0.0279 
0.0302 
0.0325 

0.0349 
0.0372 
0.0395 

0.0418 
0.0442 
0.0465 

0.0488 
0.05 I I 
0.0534 
0.0558 
0.0581 
0.0604 
0.0627 
0.0651 
0.0674 
0.0697 

0.0720 

0.0744 

! 0.0767 

' 0.0790 

0.0813 

0.0836 

0.0860 

0.0883 

0.0906 
0.0929 

0.0953 

0.0976 
0.0999 
0.1022 
o, 1046 
0.1069 
0.1092 

0.1115 

0.1 138 
0.1 162 

0.118s 

0.1208 

0.1231 

0.1254 1 

0.1278 I 

0.1301 j 
0.1324 1 
0.1348 j 
O.I37T j 
0.1394 



9 



0.0000 
0.0026 
0.0052 
0.0078 
0.0105 
0.0131 
0.0157 
0.0183 
0.0209 
0.0235 
0.0261 

0.0288 
0.0314 
0.0340 
0.0366 
0.0392 
0.0418 
0.0444 
0.0471 
0.0497 
0.0523 

0.0549 
0.0575 
0.0601 
0.0627 
0.0654 
0.0680 
0.0706 
0.0732 
0.0758 
0.0784 

0.0810 
0.0837 
0.0863 
0.0889 
0.0915 
0.0941 
0.0967 
0.0993 
0.1019 
o. 1046 

0.1072 
0.1098 
0.1 124 
0.1 150 
0.1176 
o. 1202 
0.1229 

0.1255 

0.1281 
0.1307 

0.1333 ! 

o. 1359 1 

0.1385; 

c.1411 

0.1437 1 
0.1463 i 

o. 1490 ! 
0.1516 I 

0.1542 ! 
0.1568 j 



0.0000 

0.0004 

0.0008 ! 

0.0012 I 

0.0016 

0.0020 

0.0024 

0.0029 

0.0033 

0.0037 

0.0041 

0.0045 
0.0049 
0.0053 
0.0057 
0.0061 
0.0065 
0.0069 
0.0073 
0.0077 
0.0081 

0.0086 
0.0090 
0.0094 
0.0098 
0.0102 
0.0106 

O.OIIO 

0.0114 
0.0118 
O.OI22 

0.0126 
0.0130 
0.0134 
0.0138 
0.0143 
0.0147 
0.0151 
0.0155 
0.0159 
0.0163 

0.0167 
0.0171 
0.0175 
0.0179 
0.0183 
0.0187 
0.0191 
0.0195 
0.0200 
0.0204 



0.0208 
0.0212 



0.0216 [ 


0.0220 ! 


0.0224 j 

0.0228 1 


0.0232 1 
0.0236 


0.0240 


0.0244 



13 
14 

15 
16 

17 
18 

19 

20 

21 

22 
23 

24 
25 
26 

27 
28 
29 
30 

31 
32 

33 
34 
35 
36 
37 
3S 

39 
40 

41 
42 

43 
44 

'45 
! 4 6 

47 
: 4 S 

:49 

| 5 o 

\s* 
,52 
53 
54 
55 
56 
57 
58 
59 
60 



90 


DISTANCES. 


1° 


OQ 


1 


8 


3 

2.9949 


4 


5 


6 


1 

7 

! 6.9881 


» 


9 


a 

i 1.3998 


O.O983 


1.9966 


3-9932 


4-9915 


: 5-9898 


7.9864 


S 8.9847 


01 0.9983 


1.9966 


2-9949 


j 3-9931 


4.9914 


5.9897 


: 6.9880 


1 7.9863 


8.9846 


1 * -3997 


02 0.9983 


1.9965 


2.9948 


! 3-9931 4-99!4 


5.9896 


6.9879 


i 7.9862 


i 8.9845 


I -3997 


03 0.99S3 


1.9965 


2.9948 


| 3-9931 ;i 4-99J3 


5.9896 


6.9878 


7.9861 


.. 8.9844 


!-3997 


04 0.9983 


1.9965 


2.9948 


1 3-9930 ] 4.99I3 


: 5-9895 


i 6.9878 


j 7.9860 


: 8.9843 


L3997 


05 0.99S2 


1.9965 


2.9947 


1 3.9930 4.9912 


5.9894 


! 6.9877 


1 7-9859 


8.9842 


J-3997 


06 0.9982 


1.9965 


2-9947 


3.9929 4.9912 


; 5-9894 


i 6.9876 


i 7.9S58 


; 8.9841 


L3997 


07 0.9982 


1.9964 


2.9947 


1 3.9929 4. 991 1 


j 5-9893 


6.9875 


! 7.9858 


8.9840 


I -3997 


08 0.9982 


1.9964 


2.9946 


! 3-992S 4-99 J o 


5.9S92 


6.9875 


| 7.9857 


: 8.9839 


L3997 


09 


O.9982 


1.9964 


2.9946 


3.9928 4.9910 


5.9892 


6.9874 


7.9S56 


8.9838 


L3997 


10 


O.9982 


1.9964 


2.9946 


1 3.9927 


4.9909 


5-9891 


, 6.9873 


7.9855 


8.9837 


1-3997 


II 


O.O9S2 


1.9963 


2.9945 


3-9927 j 


4.9909 


5-9890 


; 6.9872 


7-9854 


8.9836 


1-3997 


12 


O.9982 


T.9963 


2-9945 


' 3-9926 : 4.9908 


1 5.9890 


6.9871 


7.9853 


: 8.9834 


1-3997 


!3 


O.O9S1 


1.9963 


2.9944 


3.0926 ; 4.9907 


5-9889 


' 6.9870 


7-9852 


: 8.9833 


1-3997 


14 


O.99S1 


T.9963 


2.9944 


3.9925 ! 4.9907 


5.9888 


i 6.9870 


i 7-9851 


1 8.9S32 


I 1-3996 


15 


O.9981 


1.9962 


2-9944 


3.9925 4.9906 


5.9887 


1 6.9869 


I 7.9850 


i 8.9831 


! 1.3996 


16 


O.9981 


1.9962 


2-9943 


3.9924 4.9906 


5.9887 


: 6.9868 


; 7-9849 


8.9830 


i 1.3996 


17 


O.9981 


1.9962 


2-9943 


3.9924 1 4.9905 


5-9886 


6.9867 


i 7.9848 


8.9829 


1.3996 


18 


O.9981 


1.9962 


2-9943 


3.9923 4.9904 


5.9S85 


6.9S65 


7.9847 


1 8.9828 


1.3996 


19 


O.9981 


1.9962 


2.9942 


3.9923 I 4.9904 


5.98S4 


6.9S65 


7.9846 


8.9827 


1.3996 


2D 


O.9981 


1. 9961 


2.9942 


3.9922 : 4.9903 


5.9SS4 


6.9S64 


7.9845 


8.9825 


1.3996 


21 


O.9980 


1. 9961 


2.9941 


3.9922 : 4.9902 


5-9883 


6.9863 


7.9844 


8.9824 


1.3996 


22 


O.9980 


1. 9961 


2.9941 


3.9921 4.9902 


5-98S2 


6.9862 


7.9842. 


8.9823 


I-39S6 


23 O.9980 


1.9960 


2.9941 


3.9921 4.9901 


5.98S1 6.9861 


7.9841 


S.9S22 


1.3996 


24 O.99S0 


1.9960 


2.9940 


3.9920 4.9900 


5.9880 6.9860 


7.9840 


8. 9820 


I.3996 


25 O.9980 


1.9960 


2.9940 


3.9920 4.9899 


5-9879 


6.9859 


7-9S39 


8.9819 


1-3995 


26 O.9980 


1-9959 


2-9939 


3.9919 4.9899 


5.9878 


6.9858 


7.983S 


8.981S 


1-3995 


27 O.99S0 


1-9959 


2.9939 


3.9918 4.9898 


5.9S73 


6.9857 


7.9837 


8.9S16 


1-3995 


28 : O.9979 


1-9959 


2.993S 


3.9918 j 4.9897 


5-9877 


6.9856 


7.9836 


S.9S15 


1-3995 


29 O.O979 


1-9959 


2.993S 


3.9917 4.9897 


5.9876 


6.9855 


7-9S34 


S.9S14 


1-3995 


30 


O.9979 


1.9958 


2-9937 


3-9917 ; 


4.9S96 


5.9875 


6.9854 


7-9S33 


8.9812 


1-3995 


31 


O.9979 


1.9958 


2-9937 


3.9916 


4.9895 


5.9874 


6.9S53 


7.9832 


8L9811 


1-3995 


32 


O.9979 


1.9958 


2-9937 


3-99I5 


4.9894 


>9873 


6.9S52 


7-9831 


S.9S10 


1-3995 


33 


O.9979 


1-9957 


2.9936 


3.9915 .j 4.9893 


5.9872 


6.9851 


7.9S29 


S.9S0S 


1-3995 


34 


O.9979 


1-9957 


2.9936 


3.9914 4.9893 


5.9871 


6.9S50 


7.9828 


S.9S07 


1-3995 


35 0.997S 


1-9957 


2-9935 


3.9913 4.9S92 


5.9870 


6.9849 


7.9S27 


8.9805 


1-3995 


30 0.997S 


1.9956 


2-9935 


3.9913 , 4.9S9I 


5.9869 


6.9S47 


7.9S26 


8.9804 


1-3994 


37 


O.Q978 


1.9956 


2-9934 


3.9912 4.989O 


5.986S 


6.9S46 


7.9S24 


8.9802 


1-3994 


38 


O.997S 


1.9956 


2-9934 


3.99I I 4.9SS9 


5-9867 


6.9S45 


7.9823 


S.9S01 


1-3994 


39 


O.Q978 


J-9955 


2-9933 


3.991 1 4.98S9 


5.9S66 


6.9844 


7.9822 


8-9799 


1-3994 


40 


O.9978 


1-9955 


2-9933 


3.99IC . ( 4.98SS 


5-9865 


6.9S43 


7.9820 


S.979S 


r-3994 


4i 


O.9977 


L9955 


2.9932 


3.9909 ' 4.O887 


5.9S64 


6.9842 


7.9S19 


8.9796 


1-3994 


42 


O.9977 


1-9954 


2.9932 


3.9909 1 4.98S6 


5.9S63 


6.9840 


7.9818 


S-9795 


1-3994 


43 


O.9977 


1-9954 


2.9931 


3.990S i 4.98S5 


5.9S62 


6.9S39 


7.9S16 


8-9793 


1-3994 


44 


C.9977 


1-9954 


2.9931 


3.9907 J 4.98S4 


5.9S61 


6.9S3S 


7.9S15 


S.9792 


1-3994 


45 


O.9977 


1-9953 


2.9930 


3.99O7 4.9SS3 


5.9S60 


6.9S37 


7.9S13 


S.9790 


1-3994 


46: 


O.9976 


1-9953 


2.9929 


3.9906 ^ 4.0SS2 


5.9859 


6.9S35 


7.9S12 


8.97SS 


1-3993 


47 


O.9976 


1-9953 


2.9929 


3.9905 4.98S2 


5.985S 


6.9S34 


7.9S10 


S.97S7 


1-3993 


43 


O.9976 


1.9952 


2.992S 


3.99O5 4.9SSI 


5.9S57 


6.9833 


7-9809 


S.9785 


1-3993 


49 


O.9976 


1.9952 


2.992S 


5.99O4 4.OSSO 


5.9856 


6.9S32 


7.9808 


8.9784 


i-3?93 


50 


O.9976 


1.9952 


2.9927 


3.99O3 4.OS79 


5.9S55 


6.9S30 


7.9806 


8.9782 


1-3993 


5i 


O.9976 


I.995I 


2.9927 


3.99O2 4.9S7S 


5.9854 


6.9829 


7.9805 


S.97S0 


1-3993 


52 


Q-9975 


1-9951 


2.9926 


3.99O2 4-9877 


5-9 s 52 


6.982S 


7.9803 


8-9779 


1-3993 


53 1 0.9975 1 


1.9950 


2.9926 


3.990I 4-9876 


5-9S5I 


6.9S26 


7.9802 ; 


8-9777 


1-3993 


54 0.9975 1 


1.9950 


2.9925 


3.99OO 4.0S75 


5-9S50 


6.9S25 


7.9S00J 


8-9775 


1-3993 


55 0.9975 | 


1.9950 


2.9924 


3.9S99 4.9874 


5-9849 


6.9S24 


7.979S i 


8-9773 


1-3992 


56 


o.9975 ! 


1.9949 


2.9924 


3.9898 4.9S73 


5.9S4S 


6.9S22 


7-9797 


8-9772 


1.3992 


57 


0.9974 ] 


1.9949 


2.9923 


3.9S9S 4.9S72 


5.9847 


6.9S21 


7-9795 


S.9770 


1-3992 


58 


0.9974 


1.9948 


2.99 2 3 


3.9897 4.9S7I 


5-9S45 


6.9S20 , 


7-9794 


8.976S 


1.399a 


59 


0.9974 ! 


1.9948 


2.9922 


3.9896 4.9S7O j 


5.9844 


6.9S1S : 


7-9792 


S.97C6 


1-3992 


60 ! O.9974 j 


1.9948 


2.9922 


3.9S95 4.9S69 


5.9S43 


6.9S17 


7-979 1 


S.9765 


!.;;■:- 



V HEIGHTS. 91 


1 


Z 3 

1 


4 

0.0697 


5 


0.1046 


0. 1220 


8 


9 


0.0244 


1— 

[00 


0.0174 


0.0349 i 0.0523 


0.0871 


0.1394 


0.1568 


0.0177 


0.0354 1 0.0531 


0.0708 


0.0886 


0.1063 


0.1240 


0.1417 


O.I594 


; 0.0248 


iOI 


0.0180 


0.0360 0.0540 


0.0720 


0.0900 


0. 1080 


0.1260 


0.1440 


0. 1620 


0.0253 


02 


0.0183 


0.0366 1 0.0549 


0.0732 


0.0915 


0.1098 


0.1281 


0.1464 


0. 1647 


j 0.0257 


03 


0.0186 


0.0372 0.0558 


0.0743 


0.0929 


0.1115 


0.1301 


0.1487 


0. 1673 


0.0261 


04 


0.0189 


0.0378 0.0566 


0.0755 


0.0944 


0.1 133 


0.132 I 


0.1510 


0.1699 


| 0.0265 


05 


0.0192 


0.0383 1 0.0575 


0.0767 


0.0958 


0.1 150 


0.1342 


0. 1533 


0.1725 


0.0269 


06 


0.0195 


0.0389 


0.0584 


0.0778 


0.0973 


0.1 167 


0. 1362 


0.1557 


0.1751 


0.0273 


07 


0.0197 


0.0395 


0.0592 


0.0790 


0.0987 


0.1 185 


0. 1382 


0.1580 


0.1777 


0.0277 


08 


0.0200 


0.0401 


0.060 1 


0.0802 


0.1002 


0.1202 


0.1403 


0.1603 


0. 1803 


0.0281 


09 


0.0203 


0.0407 


0.0610 


0.0813 


0.1016 


0.1220 


0.1423 


0. 1626 


0. 1830 


0.0285 


10 


0.0206 


0.0412 


0.0619 


0.0825 


0.103 1 


0.1237 


0.1443 


0. 1649 


0.1856 


0.0289 


II 


0.0209 


0.0418 


0.0627 


0.0836 


0.1045 


0.1255 


0. 1464 


0.1673 


0.1882 


0.0293 


12 


0.0212 


0,0424 


0.0636 


0.0848 


0.1060 


0.1272 


0. 1484 


0. 1696 


0.1908 


0.0297 


13 


0.0215 


0.0430 


0.0645 


0.0860 


0.1075 


0. 1289 


0.1504 


0.1719 


0.1934 


0.0301 


14 


0.0218 


0.0436 


0.0653 


0.0871 


0. 1089 


0.1307 


0.1525 


0.1742 


0.1960 


0.0305 


15 


0.0221 


0.0441 


0.0662 


0.0883 


0. 1 104 


0. 1324 


0.1545 


0.1766 


0.1986 


0.0309 


l6 


0.0224 


0.0447 


0.0671 


0.0894 


O.I 1 18 


0. 1342 


0.1565 


0.1789 


0.2012 


0.0314 


17 


0.0227 


0.0453 


0.0680 


0.0906 


0.1 133 


0. 1359 


0.1586 


0.1812 


0.2039 


0.0318 


18 


0.0229 


0.0459 


0.0688 


0.0918 


0.1 147 


0.1376 


0.1606 


0.1835 


0.2065 


0.0322 


T 9 


0.0232 


0.0465 


0.0697 


0.0929 


0.1 162 


0.1394 


0. 1626 


0.1858 


0.2091 


0.0326 


20 


0.0235 


0.0470 


0.0706 


0.0941 


0.1176 


0.1411 


0. 1646 


0.1882 


0.2117 


0.0330 


21 


0.0238 


0.0476 0.0714 


0.0952 


0.1191 


0. 1429 


0.1667 


0.1905 


0.2143 


0.0334 


22 


0.0241 


0.0482 I 0.0723 


0.0964 


0.1205 


0.1446 


0.1687 


0. 1928 


0.2169 


0.0338 


23 


0.0244 


0.0488 ! 0.0732 


0.0976 


0.1220 


0.1463 


0.1707 


0.195 I 


0.2195 


0.0342 


24 


0.0247 


0.0494 


0.0740 


0.0987 


0.1234 


0.1481 


0.1728 


0.1974 


0.2221 


0.0346 


25 


0.0250 


0.0499 


0.0749 


0.0999 


0,1249 


0. 1498 


0.1748 


0.1998 


0.2247 


0.0350 


26 


0.0253 


0.0505 


0.0758 


O.IOIO 


0.1263 


0.1516 


0.1768 


0.2021 


0.2273 


0.0354 


27 


0.0256 


0.051 1 


0.0767 


0. 1022 


0.1278 


0.1533 


0.1789 


0.2044 


0.2300 


0.0358 | 


28 


0.0258 


0.0517 


0.0775 


0.1034 


0. 1292 


0.1550 


0.1809 


0.2067 


0.2326 1 


0.0362 j 


29 


0.0261 


0.0523 


0.0784 


0.1045 


0.1307 


0.1568 


0.1829 


0.2090 


0.2352 


0.0366 


30 


0.0264 


0.0528 


0.0793 


0.1057 


0.132 1 


0.1585 


0. 1849 


0.2114 


0.2378 


0.0371 


3i 


0.0267 


0.0534 


0.0801 


0.1068 


0.1336 


0. 1603 


0.1870 


0.2137 


0.2404 


0.0375 


32 


0.0270 


0.0540 


0.0810 


0. 1080 


0.1350 


0. 1620 


0. 1890 


0.2160 


0.2430 


0.0379 


33 


0.0273 


0.0546 


0.0819 


0.1092 


0.1365 


0.1637 


0.1910 


0.2183 


0.2456 


0.0383 


34 


0.0276 


0.0552 


0.0827 


0.1 103 


0.1379 


0.1655 


0.1931 


0.2206 


0.2482 


0.0387 


35 


0.0279 


0.0557 


0.0836 


0.1115 


0.1394 


0.1672 


0.195 I 


0.2230 


0.2508 


0.0391 


36 


0.0282 


0.0563 


0.0845 


0.1 126 


0. 1408 


0.1690 


0.1971 


0.2253 


0.2534 


0.0395 


37 


0.0285 


0.0569 


0.0854 


O.I 138 


0.1423 


0.1707 


0. 1992 


0.2276 


0.2561 


0.0399 


38 


0.0287 


0.0575 


0.0862 


0.1 150 


0.1437 


0.1724 


0.2012 


0.2299 


0.2587 


0.0403 


39 


0.0290 


0.0581 


0.0871 


0,1161 


0.1452 


0.1742 


0.2032 


0.2322 


0.2613 


0.0407 


40 


0.0293 


0.0586 


0.08S0 


0.1 173 


0. 1466 


0.1759 


0.2052 


0.2346 


0.2639 


0.041 1 


4i 


0.0296 


0.0592 


0.0888 


0.1 184 


0.148 I 


0.1777 


0.2073 


0.2369 


0.2665 


0.0415 


42 


0.0299 


0.0598 


0.0897 


0. 1 196 


0.1495 


0.1794 


0.2093 


0.2392 


0.2691 


0.0419 


43 


0.0302 


0.0604 


0.0906 


0.1208 


0.15 10 


0.1811 


0.2113 


0.2415 


0.2717 


0.0423 


44 


0.0305 


0.0610 


0.0914 


0.1219 


0.1524 


0.1829 


0.2134 


0.2438 


0.2743 


0.0428 


45 


0.0308 


0.0615 


0.0923 


0.1231 


0.1539 


0. 1846 


0.2154 


0.2462 


0.2769 


0.0432 


46 


0.031 1 


0.0621 


0.0932 


0.1242 


0.1553 


0. 1864 


0.2174 


0.2485 


0.2795 


0.0436 


47 


0.0314 


0.0627 


0.0941 


0.1254 


0.1568 


0.1881 


0.2195 


0.2508 


0.2822 


0.0440 


48 


0.0316 


0.0633 


0.0949 


0. 1266 


0.1582 


0.1898 


0.2215 


0.2531 


0.2848 


0.0444 


49 


0.0319 


0.0639 


0.0958 


0.1277 


0.1597 


0.1916 


0.2235 


0.2554 


0.2874 


0.0448 


50 


0.0322 


0.0644 


0.0967 


0. 1289 


0.1611 


0.1933 


0.2255 


0.2578 


0.2900 


0.0452 


5i 


0.0325 


0.0650 


0.0975 


0.1300 


0.1626 


0.195 1 


0.2276 


0.2601 


0.2926 


0.0456 1 


52 


0.0328 


0.0656 


0.0984 


0.1312 


0. 1640 


0.1968 


0.2296 


0.2624 


0.2952 | 


0.0460 


53 


0.0331 


0.0662 


0.0993 


0.1324 


0.1655 


0.1985 


0.2316 


0.2647 


0.2978 j 


0.0464 


54 


0.0334 


0.0668 


O.IOOI 


0.133s 


0.1669 


0.2003 


0.2336 


0.2670 


0.3004 


0.0468 


55 


0.0337 


0.0673 


O.IOIO 


0.1347 


0. 1684 


0.2020 


0.2357 


0.2694 


0.3030 | 


0.0472 | 


56 


0.0340 


0.0679 


0.1019 


0.1358 


0.1698 


0.2038 


0.2377 


0.2717 


0.3056 1 


0.0476 


5 Z 


0.0342 


0.0685 


0. 1027 


0.1370 


0. 1712 


0.2055 


0.2397 


0.2740 


0.3082 


0.0480 1 


58 


0.0345 


0.0691 


0. 1036 


0. 1382 


0.1727 


0.2072 


0.2418 


0.2763 


0.3109 


0.0485 I 


59 


0.0348 


0.0697 


0.1045 


0.1393 


0.1742 


0.2090 


0.2438 


0.2786 


o.3i35 


0.0489 


60 



92 DISTANCES. 


2° 


/ 
oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 

8.9765 


a 


0.9974 


1.9948 


2.9922 


3.9895 


4.9869 


5.9843 


6.9817 


7.9791 


1.3992 


OI 


0.9974 


1-9947 


2.9921 


3.9895 


4.9868 


5.9842 


6.9815 


7.9789 


8.9763 


1.3992 


02 


0.9973 


1-9947 


2.9920 


3-9894 


4.9867 


5.9841 


6.9814 


7.9787 


8.9761 


1.3992 


03 


0-9973 


1.9946 


2.9920 


3.9893 


4.9866 


5.9839 


6.9812 


7.9786 


8-9759 


1.3992 


04 


0-9973 


1.9946 


2.9919 


3.9892 


4.9865 


5.9838 


6. 981 1 


7.9784 


8-9757 


i-399 1 


05 


0-9973 


1.9946 


2.9918 


3.9891 


4.9864 


5.9837 


6.9810 


7.9782 


8-9755 


1 -399i 


c6 


o.9973 


1-9945 


2.9918 


3.9890 


4.9863 


5.9835 


6.9808 


7.9781 


8-9753 


1-399* 


07 


0.9972 


1-9945 


2.9917 


3.9889 


4.9862 


5.9834 


6.9807 


7-9779 


8-9751 


I-399I 


08 


0.9972 


1.9944 


2.9916 


3.9889 


4.9861 


5.9833 


6.9805 


7-9777 


: 8.9749 


I-399I 


09 


0.9972 


1.9944 


2.9916 


3.9888 


4.9860 


5.9832 


6.9804 


7.9776 


8-9747 


I-399I 


10 


0.9972 


1-9943 


2.9915 


3.9887 


4-9859 


5-9830 


6.9802 


7-9774 


8.9746 


I-399I 


11 


0.9972 


1-9943 


2.9915 


3.9886 


4.9858 


5.9829 


6.9801 


7.9772 


8.9744 


1.3990 


12 


0.9971 


1-9943 


2.9914 


3-9885 


4.9856 


5-9828 


6.9799 


7-9770 


8.974I 


1.3990 


13 


0.9971 


1.9942 


2.9913 


3.9884 


4.9855 


5.9826 


6.9797 


7.9768 


8-9739 


I-3990 


*4 


0.9971 


1.9942 


2.9912 


3-9883 


4.9854 


5-9825 


6.9796 


7.9767 


8-9737 


1.3990 


15 


0.9971 


1.9941 


2.9912 


3.9882 


4.9853 


5.9824 


6-9794 


7-9765 


8-9735 


1.3990 


16 


0.9970 


1.9941 


2. 991 1 


3.9881 


4.9852 


5.9822 


6.9793 


7-9763 


8-9733 


1.3990 


17 


0.9970 


1.9940 


2.9910 


3-988i 


4.9851 


5.9821 


6.9791 


7.9761 


8 9731 


1.3989 


18 


0.9970 


1.9940 


2.9910 


3.9880 


4.9850 


5.9819 


6.9789 


7-9759 


8.9729 


1.3989 


J 9 


0.9970 


1-9939 


2.9909 


3-9879 


4.9848 


5-9818 


6.9788 


7-9757 


8.9727 


1.3989 


20 


0.9969 


1-9939 


2.9908 


3.9378 


4.9847 


5-9817 


6.9786 


7-9756 


8.9725 


L3989 


21 


0.9969 


1.9938 


2.9908 


3-9877 


4.9846 


5.9815 


6.9784 


7-9754 


8.9723 


1.3989 


22 


0.9969 


1.9938 


2.9907 


3.9876 


4.9845 


5.9814 


6.9783 


7-9752 


8.972I 


1.3989 


23 


0.9969 


1-9937 


2.9906 


3.9875 


4.9844 


5.9812 


6.9781 


7-9750 


8.9718 


1.3988 


24 


0.9968 


1-9937 


2.9905 


3.9874 


4.9842 


5.9811 


6-9779 


7.9748 


8.9716 


1.3988 


25 


0.9968 


1.9936 


2.9905 


3.9873 


4.9841 


5.9809 


6.9778 


7.9746 


8.9714 


1.3988 


26 


0.9968 


1.9936 


2.9904 


3.9872 


4.9840 


5.9S08 


6.9776 


7-9744 


8.9712 


1.3988 


27 


0.9968 


1-9935 


2.9903 


3.987I 


4.9839 


5.9806 


6-9774 


7.9742 


8.97IO 


1.3988 


28 


0.9967 


1-9935 


2.9902 


3.9870 


4.9837 


5.9805 


6.9772 


7.9740 


8.9707 


1.3987 


29 


0.9967 


1-9934 


2.9902 


3-9869 


4.9836 


5.9S03 


6.9771 


7.9738 


8.9705 


1.3987 


30 


0.9967 


1-9934 


2.9901 


3.9868 


4-9835 


5.9802 


6.9769 


7.9736 


8.9703 


L3987 


3i 


0.9967 


1-9933 


2.9900 


3.9867 


4.9834 


5.9800 


6.9767 


7-9734 


8.97OI 


I.3987 


32 


0.9966 


1-9933 


2.9899 


3.9866 


4.9832 


5-9799 


6.9765 


7-9732 


8.9698 


L3987 


33 


0.9966 


1.9932 


2.9899 


3.9865 


4.9S31 


5-9797 


6.9764 


7-9730 


S.9696 


1-3987 


34 


0.9966 


1.9932 


2.9898 


3.9S64 : 


4.9830 


5.9796 


6.9762 


7.9728 


8.9694 


1.3986 


35 


0.9966 


1. 993i 


2.9897 


3.9863 : 


4.9S28 


5-9794 


6.9760 


7.9726 


8.969I 


1.3986 


36 


0.9965 


1- 993i 


2.9896 


3.9862 


4.9827 


5-9793 


6.9758 


7.9723 


8.9689 


I.39S6 


37 


0.9905 


1.9930 


2.9896 


3-9861 


4.9826 


5-9791 


6.9756 


7.9721 


8.9687 


1.3986 


38 


0.9965 


1.9930 


2.9S95 


3.9860 


4.9825 


5.9789 


6-9754 


7.9719 


8.96S4 


1.3986 


39 


0.9965 


1.9929 


2.9894 


3-9859 


4.9S23 


5.9788 


6-9753 


7.9717 


8.9682 


1.3985 


40 


0.9964 


1.9929 


2.9893 


3.9858 


4.9822 


5.9786 


6.9751 


7.9715 


8.9680 


1.3985 


4i 


0.9964 


1.992S 


2.9892 


3.9856 


4.9821 


5.9785 


6.9749 


7.97I3 


8.9677 


1.3985 


42 


0.9964 


1.9928 


2.9S91 


3-9855 


4.9819 


5.9783 


6.9747 


7.9711 


8.9674 


I.39S5 


43 


0.9964 


1.9927 


2.9891 


3-9S54 


4.9818 


5.978i 


6-9745 


7.970S 


8. 9672 


1.3985 


44 


0.9963 


1.9927 


2.9893 


3.9853 


4.9S16 


5.9780 


6-9743 


7.9706 


8.9669 


1.3984 


45 


0.9963 


1.9926 


2.98S9 


3-9852 


4.9S15 


5.9778 


6.9741 


7-97°4 


8.9667 


I.3984 


46 


0.9963 


1-9925 


2.9888 


3-9851 


4-9814 


5.9776 


6-9739 


7.9702 


8.9664 


1.3984 


47 


0.9962 


I.9925 


2.9S87 


3-9850 


4.9812 


5-9775 


6-9737 


7.9700 


S.9662 


1.3984 


48 


0.9962 


1.9924 


2.9886 


3-9S49 


4.9811 


5-9773 


6-9735 


7.9697 


S.9659 


1-3984 


49 


0.9962 


1.9924 


2.9SS6 


3.9S48 


4.9809 


5-9771 


6-9733 


7-9695 


8.9657 


1.3983 


50 


0.9962 


I-9923 


2.9885 


3.9846 


4.9S0S 


5-977Q 


6.9731 


7-9693 


8.9654 


1.3983 


5i 


0.9961 


1.9923 


2.9884 


3-9S45 


4-9807 


5.9768 


6.9729 


7.9690 


8.9652 


1.3983 


52 


0.9961 


1.9922 


2.9SS3 


3.9844 : 4.9S05 


5.9766 


6.9727 


7.96SS 


S.9649 


1.3983 


53 


0.9961 


1. 9921 


2.9SS2 


3.9843 | 4.9S04 


59764 


6.9725 


7.96S6 


S.9646 


1.3983 


54 


0.9960 


1. 992 1 


2.9881 


3.9S42 | 4.9S02 


5.9763 


6.9723 


7.9683 


S.9644 


1.39S2 


55 


0.9960 


1.9920 


2.9880 


3.9841 4.9S01 


5.976i 


6.9721 


7.96S1 


S.9641 


1.3982 


56 


0.9960 


1.9920 


2.9879 


3-9S39 


4-9799 


5-9759 


6.9719 


7.9679 


S.963S 


1.3982 


57 


0.9960 


1. 9919 


2.9879 


3.983S 1 


4-979S 


5-9757 


6.9717 


7.9676 


S.9636; 


1.3082 


58 


0.9959 


1. 9918 


2.9878 


3-9837 


4.9796 


5.9756 


6-9715 


7-9674 


8.9633 


1.39S1 


59 


0.9959 


1.9918 


2.9877 


3.9S36 


4-9795 


5-9754 


6.9713 


7.9672 


S.9631 


1.39S1 


60 


0.9959 


1.9917 


2.9876 


3.9835 1 


4-9793 


5-9752 J 


6.971 1 ! 


7.9669 


8.962S 


I-39SI 



2° HEIGHTS. 93 


1 


3 


3 


4 


5 


6 


7 


8 


9 


b 


00 


0.0348 


0.0697 


0.1045 


O.I393 


0.1742 


0.2090 


c.2438 


0.2786 


0.3I35 


0.0489 


0.0351 


0.0702 


0.1054 


0.1405 


0.1756 


0.2107 


0.2458 


0.2810 


0.3T61 


0.0493 


01 


0-0354 


0.0708 


0.1062 


0.1416 


0. 1771 


0.2125 


0.2479 


0.2833 


0.3187 


0.0497 


02 


0.0357 


0.0714 


0.1071 


0.1428 


0.1785 


0.2142 


0.2499 


0.2856 


0.3213 


0.0501 


03 


0.0360 


0.0720 


0. 1080 


0.1440 


0.1800 


0.2159 


0.2519 


0.2879 


0.3239 


0.0505 


04 


0.0363 


0.0726 


0.1088 


0.145 1 


0.1814 


0.2177 


0.2540 


0.2902 


0.3265 


0.0509 


05 


0.0366 


0.0731 


0.1097 


0.1463 


0.1828 


0.2194 


0.2560 


0.2926 


0.3291 


0.0513 


06 


0.0369 


0.0737 


0.1 106 


0.1474 


0.1843 


0.2212 


0.2580 


0.2949 


o.33i7 


0.0517 


07 


0.0371 


0.0743 


0.1114 


0. i486 


0.1857 


0.2229 


0.2600 


0.2972 


o.3343 


0.0521 


08 


0.0374 


0.0749 


0.1 123 


0. 1498 


0.1872 


0.2246 


0.2621 


0.2995 


0.3370 


0.0525 


09 


0.0377 


0.0755 


0.1 132 


0.1509 


0.1S86 


0.2264 


0.2641 


0.3018 


0.3396 


0.0529 


10 


0.0380 


0.0760 


0.1141 


0.1521 


0.1901 


0.2281 


0.2661 


0.3042 


0.3422 


0.0533 


11 


0.0383 


0.0766 


0.1 149 


0.1532 


0.1915 


0.2299 


0.2682 


0.3065 


0.3448 


0.0537 


12 


0.0386 


0.0772 


0.1 158 


0.1544 


0.1930 


0.2316 


0.2702 


0.3088 


o.3474 


0.0541 


13 


0.0389 


0.0778 


0.1167 


0.1556 


0.1944 


0.2333 


0.2722 


0.3111 


0.3500 


0.0546 


14 


0.0392 


0.0783 


0.1175 


0.1567 


0.1959 


0.2350 


0.2742 


o.3i34 


0.3526 


0.0550 


15 


0.0395 


0.0789 


0.1 184 


0.1578 


O.I973 


0.2368 


0.2762 


o.3i57 


o.3552 


0.0554 


16 


0.0398 


0.0795 


0.1 193 


0.1590 


0.1988 


0.2385 


0.2783 


0.3180 


o.3578 


0.0558 


17 


0.0400 


0.0801 


0.1201 


0.1602 


0.2002 


0.2402 


0.2803 


0.3203 


0.3604 


0.0562 


18 


0.0403 


0.0807 


0.1210 


0.1613 


0.0017 


0.2420 


0.2823 


0.3226 


0.3630 


0.0566 


19 


0.0406 


0.0812 


0.1219 


0. 1625 


0.2031 


0.2437 


0.2843 


0.3250 


0.3656 


0.0570 


20 


0.0409 


0.0818 


0.1227 


0.1636 


0.2046 


0.2455 


0.2864 


0.3273 


0.3682 


0.0574 


21 


0.0412 


0.0824 


0.1236 


0. 1648 


0.2060 


0.2472 


0.2884 


0.3296 


0.3708 


0.0578 


22 


0.0415 


0.0830 


0.1245 


0.1660 


0.2075 


0.2489 


0.2904 


0.33I9 


0-3734 


0.0582 


23 


0.0418 


0.0836 


0.1253 


0.1671 


0.2089 


0.2507 


0.2925 


0.3342 


0.3760 


0.0586 


24 


0.0421 


0.0841 


0.1262 


0.1683 


0.2103 


0.2524 


0.2945 


0.3366 


0.3786 


0.0590 


25 


0.0424 


0.0847 


0.1271 


0.1694 


0.2118 


0.2542 


0.2965 


0.3389 


0.3812 


0.0594 


26 


0.0426 


0.0853 


0.1279 


0.1706 


0.2132 


0.2559 


0.2985 


0.3412 


0.3838 


0.0598 


27 


0.0429 


0.0859 


0.1288 


0.1718 


0.2147 


0.2576 


0.3006 


o.3435 


0.3865 


0.0602 


28 


0.0432 


0.0865 


0.1297 


0.1729 


6.2161 


0.2594 


0.3026 


0.3458 


0.3891 


0.0607 


29 


0.0435 


0.0870 


0.1306 


0.1741 


0.2176 


0.261 1 


0.3046 


0.3482 


0.39I7 


0.061 1 


30 


0.0438 


0.0876 


0.1314 


0.1752 


0.2190 


0.2629 


0.3067 


0.3505 


o.3943 


0.0615 


3i 


0.0441 


0.0882 


0.1323 


0.1764 


0.2205 


0.2646 


0.3087 


0.3528 


0.3969 


0.0619 


32 


0.0444 


0.0888 


0.1331 


O.I775 


0.2219 


0.2663 


0.3107 


o.355i 


0-3995 


0.0623 


33 


0.0447 


0.0893 


0.1340 


0.1787 


0.2234 


0.2680 


0.3127 


o.3574 


0.4021 


0.0627 


34 


0.0450 


0.0899 


0.1349 


0.1798 


0.2248 


0.2698 


0.3147 


0-3597 


0.4047 


0.0631 


35 


0.0453 


0.0905 


0.1358 


0.1810 


0.2263 


0.2715 


0.3168 


0.3620 


0.4073 


0.0635 


36 


0.0455 


0.0911 


0.1366 


0.1822 


0.2277 


0.2732 


0.3188 


0.3643 


0.4099 


0.0639 


37 


0.0458 


0.0917 


t 0.i375 


0.1833 


0.2292 


0.2750 


0.3208 


0.3666 


0.4125 


0.0643 


38 


0.0461 


0.0922 


0.1384 


0.1845 


0.2306 


0.2767 


0.3228 


0.3690 


0.4151 


0.0647 


39 


0.0464 


0.0928 


0.1392 


0.1856 


0.2321 


0.2785 


0.3249 


0.3713 


0.4177 


0.0651 


40 


0.0467 


0.0934 


0. 1401 


0.1868 


0.2335 


0.2802 


0.3269 


0.3736 


0.3203 


0.0655 


41 


0.0470 


0.0940 


0.1410 


0.1880 


0.2350 


0.2819 


0.3289 


o.3759 


0.4229 


0.0659 


42 


0.0473 


0.0946 


0.1418 


0.1891 


0.2364 


0.2837 


0.3310 


0.3782 


0.4255 


0.0664 


43 


0.0476 


0.0951 


0.1427 


0.1903 


0.2378 


0.2854 


0.3330 


0.3806 


0.4281 


0.0668 


44 


0.0479 


0.0957 


0.1436 


0.1914 


0.2393 


0.2872 


o.335o 


0.3829 


0.4307 


0.0672 


45 


0.0481 


0.0963 


0.1444 


0. 1926 


0.2407 


0.2889 


0.3370 


0.3852 


0-4333 


0.0676 


46 


0.0484 


0.0969 


O.I453 


O.I937 


0.2422 


0.2906 


0.3390 


0.3875 


o.4359 


0.0680 


47 


0.0487 


0.0974 


0. 1462 


0.1949 


0.2436 


0.2923 


0.3410 


0.3898 


0.4385 


0.0684 


48 


0.0490 


0.0980 


0. 1470 


0.1960 


0.2451 


0.2941 


o.343i 


0.3921 


0.441 1 


0.0688 


49 


0.0493 


0.0986 


0.1479 


0.1972 


0.2465 


0.2958 


o.345i 


0.3944 


0.4437 


0.0692 


50 


0.0496 


0.0992 


0.1488 


0. 1984 


0.2480 


0.2975 


o.347i 


0.3967 


0.4463 


0.0696 


5i 


0.0499 


0.0998 


0.1496 


0.1995 


0.2494 


0.2993 


0.3492 


0.3990 


0.4489 


0.0700 


52 


0.0502 


0.1003 


0.1505 


0.2007 


0.2508 


0.3010 


0.3512 


0.4014 


0.45I5 


0.0704 


53 


0.0505 


0.1009 


0.1514 


0.2018 


0.2523 


0.3028 


o.3532 


0.4037 


0.4541 


0.0708 


54 


0.0507 


0.1015 


0.1522 


0.2030 


0.2537 


0.3045 


o.3552 


0.4060 


0.4567 


0.0712 


55 


0.0510 


0.102 1 


O.I53I 


0.2042 


0.2552 


0.3062 


o.3573 


0.4083 


o.4593 


0.0716 


56 


0.0513 


0. 1026 


0.1540 


0.2053 


0.2566 


0.3079 


o.3593 


0.4106 


0.4619 


0.0721 


57 


0.0516 


0. 1032 


0.1548 


0.2064 


0.2581 


0.3097 


0.3613 


0.4129 


0.4645 


0.0725 


5S 


0.0519 


0. 1038 


O.I557 


0.2076 


0.2595 


0.3114 


0.3633 0.4152 


0.4671 


0.0729 


£ 9 


0.0522 


0.1044 


0.1566 


0.2088 


0.2610 


0.3131 


0.3653 0.4175 


0.4697 


0.0733 


60 



94 DISTANCES. 


3° 


oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 


a 


0.9959 


1.9917 


2.9876 


3-9835 


4-9793 


5-9752 


6.9711 


7.9669 


8.9628 


1.3981 


OI 


0.9958 


1.9917 


2.9875 


3-9833 


4.9792 


5-975Q 


6.9708 


7.9667 


8.9625 


1-3981 


02 


0.9958 


1. 9916 


2.9874 


3-9832 


4.9790 


5-9748 


6.9706 


7.9664 


8.9622 


1-3981 


°3 


0.9958 


I.99I5 


2.9873 


3-983I 


4.9789 


5.9746 


6.9704 


7.9662 


8.9619 


1.3980 


04 


o-9957 


I-99I5 


2.9872 


3-9830 


4.9787 


5-9744 


6.9702 


7-9659 


8.9617 


1.3980 


05 


o-9957 


I.99I4 


2.9871 


3.9828 


4-9785 


5-9743 


6.9700 


7.9657 


8.9614 


1.3980 


06 


Q-9957 


I.99I4 


2.9870 


3.9827 


4.9784 


5-9741 


6.9697 


7.9654 


8. 961 1 


1.3980 


07 


0.9956 


I-99I3 


2.9869 


3.9826 


4.9782 


5-9739 


6.9695 


7.9652 


8.9608 


1.3980 


08 


0.9956 


1. 9912 


2.9868 


3-9825 


4.9781 


5-9737 


6.9693 


7.9649 


8.9605 


J-3979 


09 


0.9956 


1. 9912 


2.9868 


3-9823 


4-9779 


5-9735 


6.9691 


7.9647 


8.9603 


!-3979 


10 


0.9956 


1.9911 


2.9867 


3.9822 


4.9778 


5-9733 


6.9689 


7.9644 


8.9600 


J-3979 


11 


0.9955 


1. 9910 


2.9866 


3.9821 


4.9776 


5-9731 


6.9686 


7.9642 


8-9597 


1-3979 


12 


0.9955 


1. 9910 


2.9865 


3.9819 


4-9774 


5-9729 


6.9684 


7-9639 


8-9594 


1-3978 


J 3 


0.9955 


1.9909 


2.9864 


3.9818 


4-9773 


5-9727 


6.9682 


7.9636 


8.9591 


1-3978 


M 


0.9954 


1.9908 


2.9863 


3-98I7 


4-9771 


5-9725 


6.9679 


7-9634 


8.9588 


1-3978 


IS 


0.9954 


1.9908 


2.9862 


3.9816 


4.9769 


5-9723 


6.9677 


7.9631 


8.9585 


1-3978 


16 


0.0954 


1.9907 


2.9861 


3.9814 


4.9768 


5-9721 


6.9675 


7.9628 


8.9582 


1-3977 


17 


o.9953 


1.9906 


2.9860 


3-98I3 


4.9768 


5.9719 


6.9673 


7.9626 


8-9579 


1-3977 


18 


o.9953 


1.9906 


2.9859 


3.9812 


4.9764 


5-9717 


6.9670 


7.9623 


8.9576 


1-3977 


*9 


0-9953 


1.9905 


2.9858 


3.9810 


4.9763 


5.9715 


6.9668 


7.9621 


8-9573 


1-3977 


20 


0.9952 


1.9904 


2.9857 


3.9809 


4.9761 


5-9713 


6.9666 


7.9618 


8.9570 


1.3976 


21 


0.9952 


1.9904 


2.9856 


3.9808 


4-9759 


5.97II 


6.9663 


7.9615 


8.9567 


1.3976 


22 


0.9952 


1.9903 


2.9855 


3.9806 


4-9758 


5-9709 


6.9661 


7.9612 


8.9564 


1.3976 


23 


0.9951 


1.9902 


2.9854 


3-9805 


4-9756 


5-9707 


6.9658 


7.9610 


8.9561 


1-3976 


2 4 


0.9951 


1.9902 


2.9853 


3-9803 


4-9754 


5.97C5 


6.9656 


7.9607 


8.9558 


1-3975 


25 


0.9951 


1. 9901 


2.9852 


3.9802 


4-9753 


5.9703 


6.9654 


7.9604 


8-9555 


1-3975 


26 


0.9950 


1.9900 


2.9850 


3.9801 


4-9751 


5-9701 


6.9651 


7.9601 


8.9551 


1-3975 


27 


0.9950 


1.9900 


2.9849 


3-9799 


4.9749 


5.9699 


6.9649 


7-9599 


8.9548 


1-3975 


28 


0.9949 


1.9899 


2.9848 


3-9798 


4-9747 


5-9697 


6.9646 


7.9596 


8-9545 


1-3974 


29 


0.9949 


1.9898 


2.9847 


3-9797 


4.9746 


5.9695 


6.9644 


7-9593 


8.9542 


1-3974 


30 


0.9949 


1.9898 


2.9846 


3-9795 


4.9744 


5.9693 


6.9641 


7-9590 


8-9539 


1-3974 


3i 


0.9948 


1.9897 


2.9845 


3-9794 


4.9742 


5.9691 


6.9639 


7-9587 


8.9536 


1-3973 


32 


0.9948 


1.9896 


2.9S44 


3-9792 


4.9740 


5.9688 


6.9636 


7-9584 


8-9533 


1-3973 


33 


0.9948 


1.9895 


2.9843 


3-9791 


4-9738 


5.9686 


6.9634 


7-9582 


8.9529 


1-3973 


34 


0.9947 


1.9895 


2.9842 


3-9789 


4-9737 


5.9684 


6.9631 


7-9579 


8.9526 


1-3973 


35 


o.9947 


1.9894 


2.9841 


3.9788 


4-9735 


5.9682 


6.9629 


7-9576 


S.9523 


1.3972 


36 


0.9947 


1.9893 


2.9840 


3-9786 


4-9733 


5.9680 


6.9626 


7-9573 


8.9519 


1.3972 


37 


0.9946 


1.9893 


2.9839 


3-9785 


4-9731 


5.9678 


6.9624 


7-9570 


8.9516 


1-3972 


38 


0.9946 


1.9892 


2.9838 


3-9784 


4.9729 


5-9675 


6.9621 


7-9567 , 


8.9513 


1-3972 


39 


0.9946 


1. 9891 


2.9837 


3.9782 


4.9728 


5-9673 


6.9619 


7-9564 


S.9510 


1 -397i 


40 


0-9945 


1.9890 


2.9835 


3-978i 


4:9726 


5.9671 


6.9616 


7-956i 


8.9506 


I-397I 


4i 


0.9945 


1.9S90 


2.9834 


3-9779 


4.9724 


5.9669 


6.9613 


7.9558 


8.9503 


i-397i 


42 


0.9944 


1.9889 


2.9833 


3-9778 


4.9722 


5.9666 


6.9611 


7-9555 


S.9500 


I-397I 


43 


0.9944 


1.98S8 


2.9832 


3-9776 


4.9720 


5.9664 


6.9608 


7-9552 


8.9496 


1 -3970 


44 


0.9944 


1.9887 


2.9831 


3-9775 


4.9718 


5.9662 


6.9605 


7-9549 


8-9493 


1-3970 


45 


0.9943 


1.9887 


2.9830 


3-9773 


4.9716 


5.9660 


6.9603 


7-9546 


8.94S9 : 


1-3970 


46 


0-9943 


1.9S86 


2.9S29 


3-9772 


4.9714 


5-9657 


6.9600 


7-9543 


8.94S6 


1.3969 


47 


0.9943 


1.9885 


2.9828 


3-9770 


4-9713 


5-9655 


6.9598 


7-9540 


8.9483 


1.3969 


48 


0.9942 


1.9884 


2.9826 


3-9769 


4.9711 


5-9653 


6-9595 


7-9537 


8-9479 


1.3969 


49 


0.9942 


1.9S84 


2.9825 


3-9767 


4.9709 


5-9651 


6.9592 


7-9534 


S.9476 


1.3969 


50 


0.9941 


1.9883 


2.9S24 


3-9765 


4.9707 


5-9648 


6.9590 


7-9531 


S.9472 


1.3968 


5i 


0.9941 


1.9882 


2.9823 


3-9764 


4-9705 


5.9646 


6.95S7 


7.9528 


8.9469 


1.3968 


52 


0.9941 


1.9881 


2.9822 


3.9762 


4-9703 


5-9643 


6.9584 


7-9525 


8.9465 


1.3968 


53 


0.9940 


1.9880 


2.9821 


3-976i 


4.9701 


5-9641 


6.95S1 


7'952i 


S.9462 


1.396S 


54 


0.9940 


1.9880 


2.9819 


3-9759 


4.9699 


5-9639 


6-9579 


7-95i8 


8.9458 


1-3967 


55 


0.9939 


1.9879 


2.9818 


3-9758 


4.9697 


5-9636 


6.9576 


7-9515 


8-9455 


1.3967 


56 


0.9939 


1.9878 


2.9817 


3-9756 


4.9695 


5-9634 


6-9573 


7-9512 


S.945I 


1.3967 


57 


0.9939 


1.9877 


2.9816 


3-9754 ! 


4.9693 


5-9632 


6.9570 


7.9509 


8-9447 


I-3967 


58 


0.9938 


1.9876 


2.9815 


3-9753 : 4-969 1 


5.9629 


6.9567 


7.9506 


8-9444 


1.3966 


59 


0.993S 


1.9876 


2.9813 


3-9751 j 


4.96S9 


5.9627 


6.9565 


7.9502 


S.9440 


L3966 


60 


0.9937 


1.9875 


2.9812 


3-9750 J 


4.96S7 


5.9624 


6.9562 


7-9499 


8.9437 


1-3966 | 



3° HEIGHTS. 95 


1 


3 


3 


4 


5 


6 


7 


8 


9 


b 


00 


0.0522 


0.1044 


0.1566 


0.2088 


0.2610 


0.3131 


0.3653 


0.4I75 


0.4697 


0.0733 


0.0525 


0.1050 


O.I574 


0.2099 


0.2624 


0.3149 


0.3674 


0.4198 


0.4723 


0.0737 


01 


0.0528 


0.1055 


0.1583 


0.2111 


0.2638 


0.3166 


0.3694 


O.4222 


0.4749 


0.0741 


02 


0.0531 


0. 1061 


0.1592 


0.2122 


0.2653 


0.3184 


0.37M 


O.4245 


o.4775 


0.0745 


03 


0.0533 


0. 1067 


0.1600 


0.2134 


0.2667 


0.3201 


o.3734 


0.4268 


0.4801 


0.0749 


04 


0.0536 


0.1073 


0.1609 


0.2145 


0.2682 


0.3218 


o.3754 


0.4291 


0.4827 


0.0753 


05 


0.0539 


0. 1078 


0.1618 


0.2157 


0.2696 


0.3235 


o.3774 


0.4314 


0.4853 


0.0757 


06 


0.0542 


0.1084 


0. 1626 


0.2168 


0.2711 


0.3253 


o.3795 


0.4337 


0.4879 


0.0761 


07 


0.0545 


0.1090 


0.1635 


0.2180 


0.2725 


0.3270 


0.3815 


0.4360 


0.4905 


0.0765 


08 


0.0548 


0.1096 


0. 1644 


0.2192 


0.2739 


0.3287 


0.3835 


O.4383 


0.4931 


0.0769 


09 


0.0551 


0.1 102 


0. 1652 


0.2293 


0.2754 


0.3305 


0.3856 


O.4406 


0.4957 


0.0773 


10 


0.0554 


0.1 107 


0.1661 


0.2215 


0.2768 


0.3322 


0.3876 


0.4430 


0.4983 


0.0777 


11 


0.0557 


0.1113 


0.1670 


0.2226 


0.2783 


0.3340 


0.3896 


0.4453 


0.5009 


0.0781 


12 


0.0559 


0.1119 


0.1678 


0.2238 


0.2797 


0.3350 


0.3916 


0.4475 


0.5035 


0.0786 


13 


0.0562 


0.1 125 


0.1687 


0.2249 


0.2812 


0.3374 


0.3936 


O.4498 


0.5061 


0.0790 


14 


0.0565 


0.1 130 


0. 1696 


0.2261 


0.2826 


o.339i 


o.3956 


O.4522 


0.5087 


0.0794 


15 


0.0568 


0.1 136 


0. 1 704 


0.2272 


0.2841 


0.3409 


o.3977 


0.4545 


o.5"3 


0.0798 


16 


0.0571 


0.1 142 


0.1713 


0.2284 


0.2855 


0.3426 


0.3997 


0.4568 


o.5i39 


0.0802 


17 


0.0574 


0.1 148 


0.1722 


0.2296 


0.2869 


0.3443 


0.4017 


O.4591 


0.5165 


0.0806 


18 


0.0577 


0.1 154 


0.1730 


0.2307 


0.2884 


0.3461 


0.4038 


0.4614 


0.5191 


0.0810 


19 


0.0580 


0.1 159 


0.1739 


0.2319 


0.2898 


0.3478 


0.4058 


0.4638 


0.5217 


0.0814 


20 


0.0583 


0.1 165 


0.1748 


0.2330 


0.2913 


o.3495 


0.4078 


0.4660 


0.5243 


0.0818 


21 


0.0585 


0.1171 


0.1756 


0.2342 


0.2927 


0.3512 


0.4098 


0.4683 


0.5269 


0.0822 


22 


0.0588 


0.1177 


0.1765 


0.2353 


0.2942 


0.3530 


0.41 18 


O.4706 


0.5295 


0.0826 


23 


0.0591 


0.1 182 


O.I774 


0.2365 


0.2956 


o.3547 


0.4138 


0.4730 


0.5321 


0.0830 


24 


0.0594 


0.1 188 


0.1782 


0.2376 


0.2971 


0.3565 


0.4159 


0.4753 


o.5347 


0.0834 


25 


0.0597 


0.1 194 


0.1 791 


0.2388 


0.2985 


0.3582 


0.4179 


0.4776 


0.5373 


0.0838 


26 


0.0600 


0.1200 


0.1799 


0.2399 


0.2999 


0.3599 


0.4199 


0.4799 


o.5399 


0.0842 


27 


0.0603 


0.1205 


0.1808 


0.241 1 


0.3014 


0.3616 


0.4219 


O.4822 


0.5425 


0.0847 


28 


0.0606 


0.1211 


0.1817 


0.2422 


0.3028 


0.3634 


0.4239 


0.4845 


0-545 1 


0.0851 


29 


0.0608 


0.1217 


0.1825 


0.2434 


0.3042 


0.3651 


0.4259 


0.4868 


o.5477 


0.0855 


30 


0.061 1 


0.1223 


0.1834 


0.2446 


0.3057 


0.3668 


0.4280 


0.4891 


0.5503 


0.0859 


3i 


0.0614 


0.1229 


0.1843 


0.2457 


0.3071 


0.3686 


0.4300 


0.4914 


0.5529 


0.0863 


32 


0.0617 


0.1234 


0.1851 


0.2468 


0.3086 


0.3703 


0.4320 


0.4937 


o.5554 


0.0867 


33 


0.0620 


0.1240 


0.1860 


0.2480 


0.3100 


0.3720 


0.4340 


0.4960 


0.5580 


0.0871 


34 


0.0623 


0. 1246 


0.1869 


0.2492 


0.3H5 


o.3737 


0.4360 


0.4983 


0.5606 


0.0875 


35 


0.0626 


0.1252 


0.1877 


0.2503 


0.3129 


0.3755 


0.4381 


0.5006 


0.5632 


0.0879 


36 


0.0629 


0.1257 


0.1886 


0.2515 


0.3I43 


0.3772 


0.4401 


O.5030 


0.5658 


0.0883 


37 


0.0632 


0.1263 


0.1895 


0.2526 


0.3158 


0.3789 


0.4421 


0.5053 


c.5684 


0.0887 


38 


0.0634 


0. 1269 


0.1903 


0.2538 


0.3172 


0.3806 


0.4441 


0.5075 


0.5710 


0.0891 


39 


0.0637 


0.1275 


0.1912 


0.2549 


0.3187 


0.3824 


0.4461 


0.5098 


o.5736 


0.0895 


40 


0.0640 


0.1280 


0.192 1 


0.2561 


0.3201 


0.3841 


0.4481 


0.5122 


0.5762 


0.0899 


4 1 


0.0643 


0.1286 


0. 1929 


0.2572 


0.3215 


0.3859 


0.4502 


0.5I45 


0.5788 


0.0903 


42 


0.0646 


0.1292 


0.1938 


0.2584 


0.3230 


0.3876 


0.4522 


0.5168 


0.5814 


0.0908 


43 


0.0649 


0.1298 


0.1946 


0.2595 


0.3244 


0.3893 


0.4542 


0.5190 


0.5839 


0.0912 


44 


0.0652 


0.1303 


O.I955 


0.2607 


0.3259 


0.3910 


0.4562 


O.5214 


0.5865 


0.0916 


45 


0.0655 


0.1309 


0. 1964 


0.2618 


0.3273 


0.3928 


0.4582 


0.5237 


0.5891 


0.0920 


46 


0.0657 


o.i3!5 


0.1972 


0.2630 


0.3287 


0.3945 


0.4602 


0.5260 


0-59*7 


0.0924 


47 


0.0660 


0.132 1 


0.1981 


0. 2642 


0.3302 


0.3962 


0.4622 


0.5283 


o.5943 


0.0928 


48 


0.0663 


0. 1326 


0.1990 


0.2653 


0.3316 


0.3979 


0.4642 


0.5306 


0.5069 


0.0932 


49 


0.0666 


0.1332 


0. 1998 


0.2664 


o.333i 


0.3997 


0.4663 


0.5329 


0.5995 


0.0936 


50 


0.0669 


0.1338 


0.2007 


0. 2676 


o.3345 


0.4014 


0.4683 


0.5352 


0.6021 


0.0940 


5 1 


0.0672 


0.1344 


0.2016 


0.2688 


o.3359 


0.4031 


0.4703 


0.5375 


0.6047 


0.0944 


52 


0.0675 


0.1349 


0.2024 


0.2699 


o.3374 


0.4048 


0.4723 


0.5398 


0.6073 


0.0948 


53 


0.0678 


o.i355 


0.2033 


0.2710 


0.3388 


0.4066 


o.4743 


0.5421 


0.6099 


0.0952 


54 


0.0681 


0.1361 


0.2042 


0.2722 


0.3403 


0.4083 


0.4764 


0.5444 


0.6125 


0.0956 


55 


0.0683 


0.1367 


0.2050 


0.2734 


o.34i7 


0.4100 


0.4784 


0.5467 


0.6151 


0.0961 


56 


0.06S6 


O.I373 


0.2059 


0.2745 


o.343i 


0.41 18 


0.4804 


0.5490 


0.6177 


0.0965 


57 


0.0689 


0.1378 


0.2067 


0.2756 


0.3446 


0.4I35 


0.4824 


0.55I3 


0.6202 


0.0969 


58 


0.0692 


0.1384 


0.2076 


0.2768 


0.3460 


0.4152 


0.4844 


0.5536 


0.6228 


0.0973 


59 


0.0695 


0.1390 


0.2085 


0.2780 


0.3474 


0.4169 


0.4864 


0.5559 


0.6254 


0.0977 


60 



96 DISTANCES. 


4° 


oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 


a 


0.9937 


I-9875 


2.9812 


3.9750 


4.9687 


5.9624 


6.9562 


7-9499 


8-9437 


1.3966 


OI 


0.9937 


1.9874 


2.9811 


3.9748 


4.9685 


5.9622 


6-9559 


7.9496 


8-9433 


1.3966 


02 


o.9937 


1.9873 


2.9810 


3.9746 


4.9683 


5.9619 


6.9556 


7-9493 


8.9429 


1-3965 


03 


0.9936 


1.9872 


2.9809 


3-9745 


4.9681 


5-96I7 


6-9553 


7.9489 


8.9426 


1-3965 


04 


0.9936 


1.9872 


2.9807 


3-9743 


4.9679 


5.9615 


6.9550 


7.9486 


8.9422 


1.3965 


05 


°-9935 


1.9871 


2.9806 


3.9741 


4.9677 


5.9612 


6-9547 


7-9483 


8.9418 


1.3965 


06 


Q-9935 


1.9S70 


2.9805 


3.9740 


4.9675 


5.9610 


6-9545 


7-9479 


8.9414 


1.3964 


07 


o.9935 


1.9869 


2.9804 


3.9738 


4.9673 


5.9607 


6.9542 


7.9476 


8.941 1 


1.3964 


08 


0-9934 


1.986S 


2.9802 


3-9730 


4.9671 


5-9605 


6-9539 


7-9473 


8.9407 


1.3964 


09 


0.9934 


1.9867 


2.9801 


3-9735 


4.9668 


5.9602 


6.9536 


7.9470 


8.9403 


1-3963 


10 


o.9933 


1.9867 


2.9800 


3-9734 


4.9666 


5.9600 


6-9533 


7.9466 


8.9400 


1-3963 


II 


o.9933 


1.9S66 


2.9799 


3.9731 


4.9664 


5-9597 


6.9530 


7.9463 


8.9396 


1.3963 


12 


0.9932 


1.9865 


2.9797 


3-9730 


4.9662 


5-9595 


6.9527 


7-9459 


8.9392 


L3963 


13 


0.9932 


1.9864 


2.9796 


3.9728 


4.9660 


5.9592 


6.9524 


7-9456 


8.9388 


1.3962 


M 


0.9932 


1.9863 


2-9795 


3.9726 


4.9658 


5-9589 


6.9521 


7-9452 


8.9384 


1.3962 


*5 


0.9931 


1.9862 


2-9793 


3.9725 


4.9656 


5.9587 


6.9518 


7-9449 


8.9380 


1.3962 


16 


0.9931 


1. 9861 


2.9792 


3.9723 


4.9654 


5-9584 


6.9515 


7.9446 


8.9376 


1.3962 


J 7 


0.9930 


1. 9861 


2.9791 


3.9721 


4-9651 


5.9582 


6.9512 


7.9442 


8-9373 


1. 3961 


18 


0.9930 


1.9860 


2.9790 


3.9719 


4.9649 


5-9579 


6.9509 


7-9439 


8.9369 


1.3961 


19 


0.9929 


1.9859 


2.9788 


3.9718 


4.9647 


5-9577 


6.9506 


7-9435 


8.9365 


1.3961 


20 


0.9929 


1.9858 


2.9787 


3.9716 


4.9645 


5-9574 


6.9503 


7-9432 


8.9361 


1.3960 


21 


0.9929 


1 -9857 


2.9786 


3.9714 


4-9643 


5-9571 


6.9500 


7.9428 


8.9357 


1.3960 


22 


0.9928 


1.9856 


2.9784 


3-97 12 


4.9641 


5-9569 


6.9497 


7-9425 


8.9353 


1.3960 


23 


0.9928 


L9855 


2.9783 


3-97H 


4.9638 


5.9566 


6-9494 


7.9421 


8.9349 


1-3959 


24 


0.9927 


1.9854 


2.9782 


3.9709 


4.9636 


5.9563 


6.9490 


7.9418 


8-9345 


1-3959 


25 


0.9927 


1.9854 


2.9780 


3.9707 


4-9634 


5.956i 


6.9487 


7.94I4 


8.9341 


1-3969 


26 


0.9926 


L9853 


2.9779 


3.9705 


4.9632 


5.9558 


6.9484 


7.9410 


8.9337 


1.3958 


27 


0.9926 


1.9852 


2.9778 


3.9703 


4.9629 


5-9555 


6.9481 


7.9407 


8-9333 


I.3958 


28 


0.9925 


1.9851 


2.9776 


3.9702 


4.9627 


5-9553 


6.9478 


7-9403 


8.9329 


I-3958 


29 


0.9925 


1.9850 


2-9775 


3.9700 


4.9625 


5-955Q 


6-9475 


7.9400 


8.9325 


1.3958 


30 


0.9925 


1.9849 


2.9774 


3.9698 


4.9623 


5-9547 


6.0472 


7.9396 


8.9321 


1-3957 


3i 


0.9924 


1.9S48 


2.9772 


3.9696 


4.9620 


5-9544 


6.9468 


7-9393 


8.9317 


L3957 


32 


0.9924 


1.9847 


2.9771 


3-9694 


4.9618 


5-9542 


6.9465 


7.9389 


8.9312 


1-3957 


33 


0.9923 


1.9846 


2.9769 


3-9693 


4.9616 


5-9539 


6.9462 


7.9385 


8.9308 


1-3956 


34 


0.9923 


L9845 


2.9768 


3.9691 


4.9613 


5-9536 


6.9459 


7.938i 


8.9304 


1-3956 


35 


0.9922 


1.9844 


2.9767 


3.9689 


4.961 1 


5-9533 


6.9456 


7-9378 


8.9300 


1-3956 


36 


0.9922 


1.9844 


2.9765 


3.9687 


4.9609 


5-9531 


6.9452 


7-9374 


8.9296 


1-3955 


37 


0.9921 


1.9843 


2.9764 


3.9685 


4.9606 


5-9528 


6-9449 


7-9370 


8.9292 


1-3955 


38 


0.9921 


1.9842 


2.9762 


3.96S3 


4.9604 


5-9525 


6.9446 


7.9367 


8.9287 


1-3955 


39 


0.9920 


1. 9841 


2.9761 


3.9681 


4.9602 


5-9522 


6-9443 


7.9363 


8.9283 


1-3954 


40 


0.9920 


1.9840 


2.9760 


3.9680 


4.9600 


5.9519 


6-9439 


7-9359 


8.9279 


1-3954 


41 


0.9919 


1.9839 


2.9758 


3.9678 


4-9597 


5-9517 


6.9436 


7-9355 


S.9275 


L3954 


42 


0.9919 


1.9838 


2-9757 


3.9676 


4-9595 


5-9514 


6-9433 


7-9352 


8.9270 


1-3953 


43 


0.9918 


1.9837 


2-9755 


3 9674 


4.9592 


5-95U 


6.9429 


7-9348 


S.9266 


1-3953 


44 


0.9918 


1.9S36 


2-9754 


3.9672 


4.9590 


5-95oS 


6.9426 


7-9344 


8.9262 


1-3953 


45 


0.9918 


I.9835 


2-9753 


3.9670 


4.9588 


5.9505 


6.9423 


7-9340 


S.9258 


1-3952 


46 


0.9917 


1.9834 


2.9751 


3.9668 


4.95S5 


5-9502 


6.9419 


7-9336 


8.9253 


1.3952 


47 


0.9917 


1.9833 


2.9750 


3.9666 


4-9583 


5-9499 


6.9416 


7-9332 


S.9249 ! 


1-3952 


48 


0.9916 


1.9832 


2.9748 


3.9664 


4.9580 


5-9496 


6.9412 


7-9329 


8.9245 


I-395I 


49 


0.9916 


1.9831 


2-9747 


3.9662 


4.9578 


5-9494 


6.9409 


7-9325 


8.9240 


I-395I 


50 


0.9915 


1.9830 


2-9745 


3.9660 


4-9576 


5-949 1 


6.9406 


7.9321 


8.9236 


I-395I 


5i 


0.9915 


1.9829 


2.9744 


3.9658 


4-9573 


5.94SS 


6.9402 


7-93J7 


8.9231 


I.3950 


52 


0.9914 


1.9S28 


2.9742 


3-9656 


4-9571 


5.9485 


6-9399 


7-9313 


8.9227 


I-3950 


53 


0.9914 


1.9827 


2.9741 


3-9654 


4.9568 


5.9482 


6.9395 


7-9309 


8.9223 


I-3950 


54 


0.9913 


1.9S26 


2-9739 


3-9653 


4.9566 


5-9479 


6.9392 


7-9305 


S.9218 


1-3949 


55 


0.9913 


1.9825 


2.9738 


3-9651 


4-9563 


5-9476 


6.93SS 


7-93 QI 


8.9214 


1.3949 


56 


0.9912 


1.9824 


2.9736 


3.9649 


4.9561 


5-9473 


6.9385 


7.9297 


S.9209 


1-3949 


57 


0.9912 


1.9823 


2-9735 


3-9647 


4-9558 


5-947° 


6.9381 


7-9293 


S.9205 


1.39*8 


58 


0.9911 


1.9S22 


2-9733 


3-9645 


4-9556 


5-9467 


6.9378 


7-9289 


S.9100 


1.3948 


59 


0.9911 


1. 9821 


2.9732 


3.9643 


4-9553 


5-9464 


6-9375 


7-92S5 


8.9196 


1-3948 


60 


0.9910 


1.9820 


2.9730 


3.9641 


4-9551 


5.946i 


6.9371 


7.92S1 


S.9191 


1-3947 



4° HEIGHTS. 97 


1 


2 


3 


4 


5 


6 


7 


8 


9 


b 


CO 


0.0695 


0.1390 


0.2085 


0.2780 


o.3474 


0.4169 


0.4864 


o.5559 


0.6254 


0.0977 


0.0698 


0.1396 


0.2093 


0.2791 


0.3489 


0.4187 


0.4884 


0.5582 


0.6280 


0.0981 


OI 


0.0701 


0.1401 


0.2102 


0.2802 


0.35O3 


0.4204 


0.4904 


0.5605 


0.6306 


0.0985 


C2 


0.0704 


0. 1407 


0.2111 


0.2814 


0.3518 


0.4221 


0.4925 


0.5628 


0.6332 


c.0989 


03 


0.0706 


0.1413 


0.2119 


0.2826 


o.3532 


0.4238 


0.4945 


0.5651 


0.6358 


0.0993 


04 


0.0709 


0.1419 


0.2128 


0.2837 


0.3546 


0.4256 


0.4965 


0.5674 


0.6384 


0.0997 


05 


0.0712 


0. 1424 


0.2136 


0.2848 


0.3561 


0.4273 


0.4985 


0.5697 


0.6409 


O.IOOI 


06 


0.0715 


0.1430 


0.2145 


0.2860 


0.3575 


0.4290 


0.5005 


0.5720 


0.6435 


0.1005 


07 


0.0718 


0.1436 


0.2154 


0.2872 


0.3589 


0.4307 


0.5025 


0-5743 


0.6461 


0.1009 


08 


0.0721 


0.1442 


0.2162 


0.2883 


0.3604 


0.4325 


0.5045 


0.5766 


0.6487 


0.1013 


09 


0.0724 


0.1447 


0.2171 


0.2894 


0.3618 


0.4342 


0.5065 


0.5789 


0.6513 


0.1017 


IO 


0.0727 


O.I453 


0.2180 


0.2906 


0.3633 


o.4359 


0.5086 


0.5812 


0.6539 


0.I02I 


II 


0.0729 


0.1459 


0.2188 


0.2918 


0.3647 


0.4376 


0.5106 


0.5835 


0.6565 


O.IO25 


12 


0.0732 


0.1465 


C.2197 


0.2929 


0.3661 


0-4394 


0.5126 


0.5858 


0.6591 . 


O. IO29 


13 


0.0735 


0.1470 


0.2205 


0.2940 


0.3676 


0.441 1 


0.5146 


0.5881 


0.6616 


O.IO33 


14 


0.0738 


0.1476 


0.2214 


0.2952 


0.3690 


0.4428 


0.5166 


0.5904 


0.6642 


O.IO37 


15 


0.0741 


0. 1482 


0.2223 


0.2964 


0.3704 


0-4445 


0.5186 


0.5927 


0.6668 


O. IO4 1 


l6 


0.0744 


0.1488 


0.2231 


0.2975 


0.37I9 


0.4463 


0.5206 


c.5950 


0.6694 


O. IO46 


17 


0.0747 


0. 1493 


0.2240 


0.2986 


o.3733 


0.4480 


0.5226 


o.5973 


0.6720 


O. IO5O 


l8 


0.0749 


0.1499 


0.2248 


0.2998 


0.3747 


0.4497 


0.5246 


0.5996 


0.6746 


O. IO54 


19 


0.0752 


0.1505 


0.2257 


0.3010 


0.3762 


0.4514 


0.5266 


0.6019 


0.6772 


O.IO58 


20 


0.0755 


0.1510 


0.2266 


0.3021 


o.3776 


0.453I 


0.5286 


0.6042 


0.6797 


O.I062 


21 


0.0758 


0.1516 


0.2274 


0.3032 


0.3791 


0.4549 


0.5307 


0.6065 


06823 


O.I066 


22 


0.0761 


0.1522 


0.2283 


0.3044 


0.3805 


c.4566 


0.5327 


o.6c88 


0.6849 


O. IO7O 


23 


0.0764 


0.1528 


0.2292 


0.3056 


0.3819 


0.4583 


o.5347 


0.61 1 1 


0.6875 


O.IO74 


24 


0.0767 


O.I533 


0.2300 


0.3067 


0.3834 


0.4600 


0.5367 


0.6134 


0.6900 


O.IO78 


25 


0.0770 


O.I539 


0.2309 


0.3078 


0.3848 


0.4618 


0.5387 


0.6157 


0,6926 


O.I082 


26 


0.0772 


O.I545 


0.2317 


0.3090 


0.3862 


0.4635 


0.5407 


0.6180 


0.6952 


O.I086 


27 


0.0775 


O.I55I 


0.2326 


0.3101 


0.3877 


0.4652 


0.5427 


0.6203 


0.6978 


O.IO9O 


28 


0.0778 


0.1556 


0.2335 


0.3113 


0.3891 


0.4669 


0.5447 


0.6226 


0.7004 


O.IO94 


29 


0.0781 


0.1562 


0.2343 


0.3124 


0.3905 


0.4687 


0.5467 


0.6249 


0.7030 


O.IC98 


30 


0.0784 


0.1568 


0.2352 


0.3136 


0.3920 


0.4703 


0.5487 


0.6271 


0.7055 


0.1 102 


31 


0.0787 


O.I574 


0.2360 


o.3i47 


o.3934 


0.4721 


0.5508 


0.6294 


0.7081 


0.1107 


32 


0.0790 


O.I579 


0.2369 


0.3I59 


0.3948 


0.4738 


0.5528 


0.6318 


0.7107 


O.IIII 


33 


0.0793 


0.1585 


0.2378 


0.3170 


0.3963 


o.4755 


o.5548 


0.6340 


o.7i33 


0.1115 


34 


0.0795 


0.1591 


0.2386 


0.3182 


0.3977 


0.4772 


0.5568 


0.6363 


o.7i59 


0.1119 


35 


0.0798 


O.I597 


0-2395 


o.3i93 


0.3991 


0.4790 


0.5588 


0.6386 


0.7185 


0.1 123 


36 


0.0801 


o.i6c2 


0.2403 


0.3204 


0.4007 


0.4807 


0.5608 


0.6409 


0.7210 


0.1127 


37 


0.0804 


0.1608 


0.2412 


0.3216 


0.4020 


0.4824 


0.5628 


0.6432 


0.7236 


0.1 131 


38 


0.0807 


0.1614 


0.2421 


0.3228 


0.4034 


0.4841 


0.5648 


0.6455 


0. 7262 


c.1135 


39 


0.0810 


0. 1620 


0.2429 


0.3239 


0.4049 


0.4859 


0.5668 


0.6478 


0.7288 


0.1139 


40 


0.0813 


0.1625 


0.2438 


0.3250 


0.4063 


0.4876 


0.5688 


0.6503 


0.73I3 


0.1 143 


4 1 


0.0815 


0.1631 


0.2446 


0.3262 


0.4077 


0.4893 


0.5708 


0.6524 


o.7339 


0.1147 


42 


0.0818 


0.1637 


o.2455 


o.3273 


0.4092 


0.4910 


0.5728 


0.6546 


0.7365 


0.1151 


43 


0.0821 


0.1642 


0.2464 


0.3285 


0.4106 


0.4927 


0.5748 


0.6570 


0.7391 


0.1 155 


44 


0.0824 


0.1648 


0.2472 


0.3296 


0.4120 


0.4945 


0.5768 


0.6593 


0.7417 


0.1 159 


45 


0.0S27 


0.1654 


0.2481 


0.3308 


0.4I35 


0.4961 


0.5788 


0.6615 


0.7442 


0.1 163 


46 


0.0830 


0.1660 


0.2489 


0.33I9 


0.4149 


o.4979 


0.5809 


0.6638 


0.7468 


0.1167 


47 


0.0833 


0.1665 


0.2498 


o.333i 


0.4163 


0.4996 


0.5829 


0.6662 


0.7494 


0.1171 


48 


0.0836 


0.1671 


0.2507 


0.3342 


0.4178 


0.5013 


0.5849 


0.6684 


0.7520 


0.1 1 76 


49 


0.0838 


0.1677 


0.2515 


o.3354 


0.4192 


0.5030 


0.5869 


0.6707 


0.7546 


0.1 1 80 


5o 


0.0841 


0.1683 


0.2524 


0.3365 


0.4206 


0.5048 


0.5889 


0.6730 


0.7572 


0.1184 


5i 


0.0844 


0.1688 


0.2532 


0.3376 


0.4221 


0.5065 


0.5909 


0.6753 


0-7597 


0.1188 


52 


0.0847 


0.1694 


0.2541 


0.3388 


0.4235 


0.5082 


0.5929 


0.6776 


0.7623 


0.1 192 


53 


0.0850 


0.1700 


0.2549 


0.3399 


0.4249 


0.5099 


0.5949 


0.6798 


0.7648 


0.1 196 


54 


0.0853 


0.1705 


0.2558 


0.3411 


0.4264 


0.5 1 16 


0.5969 


0.6822 


0.7674 


0. 1200 


55 


0.0856 


0.1711 


0.2567 


0.3422 


0.4278 


o.5i34 


0.5989 


0.6845 


0.7700 


0.1204 


56 


0.0858 


0.1717 


0.2575 


0.3434 


0.4292 


0.5150 


0.6009 


0.6867 


0.7726 


0.1208 


57 


0.0861 


0.1723 


0.2584 


o.3445 


0.4306 


0.5168 


0.6029 


0.6890 


0.7752 


O.I2I2 


58 


0.0864 


0.1728 


0.2593 


o.3457 


0.4321 


0.5185 


0.6049 


0.6914 


0.7778 


O.I2l6 


59 


0.0867 


O.I734 


0.2601 


0.3468 


o.4335 


0.5202 


0.6069 


0.6936 


0.7803 


c.i 220 


60 



43 



98 DISTANCES. 


5° 


oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 

8.9191 


a 


c.9910 


1.9820 


2.9730 


3.9641 


4-9551 


5-946i 


6.9371 


7.9281 


1-3947 


Ol 


c.9910 


1. 9819 


2.9729 


3.9639 


4.9548 


5-9458 


6.9367 


7.9277 


8.9187 


I -3947 


02 


0.9909 


1. 9818 


2.9727 


3.9636 


4.9546 


5-9455 


6.9364 


7.9273 


8.9182 


1.3946 


03 


0.9909 


1.9817 


2.9726 


3-9634 


4-9543 


5-9452 


6.9360 


7.9269 


8.9177 


1.3946 


04 


0.9908 


1.9816 


2.9724 


3.9632 


4-9541 


5-9449 


6-9357 


7.9265 


8.9!73 


1-3946 


05 


0.9908 


1.9815 


2.9723 


3-9630 


4-9538 


5.9446 


6-9353 


7.9261 


8.9168 


1-3945 


06 


0.9907 


1. 9814 


2.9721 


3.9628 


4-9535 


5.9442 


6-9349 


7.9257 


8.9164 


1-3945 


07 


0.9907 


1.9813 


2.9720 


3.9626 


4-9533 


5-9439 


6.9346 


7.9252 


8.9159 


1-3944 


08 


0.9906 


1. 9812 


2.9718 


3.9624 


4-953o 


5.9436 


6.9342 


7.9248 


8.9154 


J-3944 


09 


0.9906 


1.9811 


2.9717 


3.9622 


4.9528 


5-9433 


6-9339 


7.9244 


8.9150 


1-3944 


10 


0.9905 


1. 9810 


2.9715 


3.9620 


4-9525 


5-9430 


6-9335 


7.9240 


8.9145 


1-3943 


II 


0.9904 


1.9809 


2.9713 


3.9618 


4.9522 


5.9427 


6.9331 


7.9236 


8.9140 


1-3943 


12 


0.9904 


1.9808 


2.9712 


3.9616 


4.9520 


5.9424 


6.9328 


7.9232 


8.9136 


1-3942 


J 3 


0.9903 


1.9807 


2.9710 


3-96i4 


4.9517 


5-9421 


6.9324 


7.9227 


8.9131 


1-3942 


*4 


0.9903 


1.9806 


2.9709 


3.9612 


4.95I5 


5.94I7 


6.9320 


7.9223 


8.9126 


1 -394i 


15 


0.9902 


1.9805 


2.9707 


3.9610 


4.9512 


5.94I4 


6.9317 


7.9219 


8.9121 


I-394I 


16 


0.9902 


1.9804 


2.9706 


3.9607 


4.9509 


5-94II 


6..93I3 


7.92I5 


8.9117 


i-394i 


17 


0.9901 


1.9803 


2.9704 


3-9605 


4.9507 


5.9408 


6.9309 


7.9211 


S.9112 


1.3940 


18 


0.9901 


1.9802 


2.9702 


3.96o3 


4.9504 


5.9405 


6.9306 


7.9206 


8.9107 


I-394Q 


19 


0.9900 


1. 9801 


2.9701 


3.9601 


4.9501 


5.9402 


6.9302 


7.9202 


8.9102 


1.3940 


2C 


0.9900 


1.9799 


2.9699 


3-9599 


4.9499 


5.9398 


6.9298 


7.9198 


8.9098 


1-3939 


21 


0.9899 


1.9798 


2.9698 


3-9597 


4.9496 


5-9395 


6.9294 


7.9193 


8.9093 


1-3939 


22 


0.9899 


1.9797 


2.9696 


3-9595 


4-9493 


5-9392 


6.9290 


7.9189 


8.9088 


1-3938 


23 


0.9898 


1.9796 


2.9694 


3-9592 


4.9490 


5-9389 


6.9287 


7.9185 


8.9083 


! 1-3938 


24 


0.9898 


1-9795 


2.9693 


3-9593 


4.9488 


5-9385 


6.9283 


7.9180 


8.9078 


: 1-3938 


2 5 


0.9897 


1.9794 


2.9691 


3-9588 


4.9485 


5-9382 


6.9279 


7.9176 


8.9073 


1-3937 


26 


0.9896 


1-9793 


2.9689 


3.9586 


4.9482 


5-9379 


6.9275 


7.9172 


8.9068 


, 1-3937 


27 


0.9896 


1.9792 


2.9688 


3-9584 


4.9480 


5-9375 


6.9271 


7.9167 


8.9063' 


1-3936 


28 


0.9895 


1.9791 


2.9686 


3.958i 


4-9477 


5-9372 


6.9268 


7-9*63 


8.9058 ■ 


' 1-3936 


29 


0.9895 


1.9790 


2.96S4 


3-9579 


4.9474 


5-9369 


6.9264 


7.9I59 


8.9053 


I-3936 


30 


0.9S94 


1.9789 


2.9683 


3-9577 


4-9471 


5-9366 


6.9260 


7.9I54 


8.904S 


1-3935 


3i 


0.9894 


1.9787 


2.9681 


3-9575 


4.9469 


5-9362 


6.9256 


7-9 T 5o 


S.9043 


1-3935 


32 


0.9893 


1.9786 


2.9679 


3-9573 


4.9466 


5-9359 


6.9252 


7.9145 


8.903S 


1-3934 


33 


0.9893 


1.9783 


2.9678 


3-9570 


4-9463 


5-9355 


6.9248 


7-9141 


8.9033 


L3934 


34 


0.9892 


1.9784 


2.9676 


3-9568 


4.9460 


5-9352 


6.9244 


7-9136 


8.9028 


1-3934 


35 


0.9891 


I-9783 


2.9674 


3-9566 


4-9457 


5-9349 


6.9240 


7-9132 


S.9023 


1-3933 


36 


0.9891 


1.9782 


2.9673 


3-9564 


4-9454 


5-9345 


6.9236 


7.9127 


S.901S 


1-3933 


37 


0.9890 


1.9781 


2.9671 


3-956i 


4-9452 


5-9342 


6.9232 


7-9123 


8.9013 


1.3932 


33 


0.9890 


1.9780 


2.9669 


3-9559 


4.9449 


5-9339 


6.9228 


7.9118 


8.9008 ! 


1-3932 


39 


0.9889 


1.9778 


2.9668 


3-9557 


4.9446 


5-9335 


6.9224 


7.9114 


8.9003 


1-3932 


40 


0.9889 


1-9777 


2.9666 


3-9555 


4-9443 


5-9332 


6.922c 


7-9109 


8.899S 


I-393I 


4i 


0.9S88 


1.9776 


2.9664 


3-9552 


4.9440 


5-9328 


6.9216 


7.9104 


S.8993 


i- 393i 


42 


0.98S7 


1-9775 


2.9662 


3-955Q 


4-9437 


5-9325 


6.9212 


7.9100 


S.89S7 


1 -3930 


43 


0.9S87 


1-9774 


2.9661 


3.9548 


4-9435 


5-9321 


6.9208 


7.9095 


8.8982 | 


1-3930 


44 


0.9S86 


1-9773 


2.9659 


3-9545 


4-9432 


5.93i8 


6.9204 


7.9091 


8.S977 


1 -393Q 


45 


0.9886 


1.9772 


2.9657 


3-9543 


4.9429 


5-9315 


6.9200 


7.90S6 


8.8972 


1-3929 


46 


0.98S5 


1.9770 


2.9656 


3-9541 


4.9426 


5.93i 1 


6.9196 


7.90S1 


8.8967 


I-3929 


47 


0.9885 


1.9769 


2.9654 


3-953S 


4-9423 


5-93oS 


6.9192 


7.9077 


S.S961 


1.392S 


48 


0.9SS4 


1.9768 


2.9652 


3.9536 


4.9420 


5.93C4 


6.91S8 


7.9072 


S.S956 


1.392S 


49 


0.98S3 


1.9767 


2.9650 


3-9534 


4.9417 


5-9300 


6.91S4 


7.9067 


8.S951 


1.3928 


50 


0.98S3 


1.9766 


2.9649 


3-9531 


4.9414 


5.9297 


6.91S0 


7.9063 


S.S946 


I-3927 


5i 


0.9882 


I-9765 


2.9647 


3-9529 


4.9411 


5.9294 


6.9176 


7.905S 


S.S940 


1.3927 


52 


0.9882 


I-9763 


2.9645 


3-9527 


4.940S 


5.9290 


6.9172 


7-9053 


S.S935 


1.3926 


53 


0.9881 


1.9762 


2.9643 


3-9524 


4-9405 


5.92S6 


6.9167 


7-9048 


S.Sg^o 


1.3926 


54 


0.9880 


1.9761 


2.9641 


3-9522 


4.9402 


5-92S3 


6.9163 


7.9044 


8.8924 


1.3926 


55 


0.9880 


1.9760 


2.9640 


3-9519 


4-9399 


5-9279 


6.9159 


7-9039 


S.S919 


1-3925 


56 


0.9879 


1-9759 


2.963S 


3-9517 


4-9396 


5-9276 


6.9155 


7-9034 


8.8913 


L3925 


57 


0.9879 


1-9757 


2.9636 


3-95I5 


4-9393 


5-9272 


6.9151 


7.9029 


S.S90S 


1.3924 


58 


0.9878 


I-9756 


2.9634 


3.9512 ; 4.9390 


5-9268 


6.9147 


7.9025 


S.S903 


1.3924 


59 


c.9877 


1-9755 


2.9632 


3-95io 


4-93S7 


5-9265 


6.9142 


7.9020 




I.3924 


60 


0.9S77 


1-9754 


2.9631 


3-95o8 


4-93S4 


5.9261 


6.913S 


7-0015 


S.SS92 ; 


I-3923 



5° HEIGHTS. 99 


1 


2 


3 


4 


5 


6 


7 


8 


9 


b 


00 


0.0867 


o.i734 


0.2601 


0.3468 


0.4335 


0.5202 


0.6069 


0.6936 


0.7803 


0.1220 


0.0870 


0.1740 


0.2610 


0.3480 


0.4349 


0.5219 


0.6089 


0.6959 


0.7829 


0.1224 


01 


0.0873 


0.T745 


0.2618 


0.3491 


0.4364 


0.5236 


0.6109 


0.6982 


0.7854 


0.1228 


02 


0.0876 


0.1751 


0.2627 


0.3502 


C.4378 


0.5254 


0.6129 


0.7005 


0.7880 


0.1232 


03 


0.0878 


o.i757 


0.2635 


0.35I4 


0.4392 


0.5271 


0.6149 


0. 7028 


0. 7906 


0.1236 


04 


0.0881 


0.1763 


0.2644 


0.3525 


0.4407 


0.5288 


0.6169 


0.7050 


0.7932 


0. 1240 


05 


0.0884 


0.1768 


0.2653 


0.3537 


0.4421 


0.5305 


0.6189 


0.7074 


0.7958 


0.1244 


06 


0.0887 


0.1774 


0.2661 


0.3548 


o.4435 


0.5322 


0.6209 


0.7096 


0.7983 


0.1248 


07 


o.oSgo 


0.1780 


0.2670 


0.3560 


0.4450 


o.5539 


0.6229 


0.71 19 


0.8009 


0.1253 


08 


0.0893 


0.1786 


0.2678 


0.357I 


0.4464 


o.5357 


0.6249 


0.7142 


0.8035 


0.1257 


09 


0.0896 


0.1 791 


0.2687 


O.3582 


0.4478 


o.5374 


0.6269 


0.7165 


0.8060 


0.1261 


10 


0.0898 


0.1797 


0.2695 


0-3594 


0.4492 


o.539i 


0.6289 


0.7188 


0.8086 


0.1265 


11 


0.0901 


0.1803 


0.2704 


O.3605 


0.4507 


0.5408 


0.6309 


0.7211 


c.8112 


0.1269 


12 


0.0904 


0.1808 


0.2713 


0.3617 


0.4521 


0.5425 


0.6329 


0.7234 


0.8138 


0.1273 


13 


0.0907 


0.1814 


0.2721 


0.3628 


o.4535 


0.5442 


0.6349 


0.7256 


0.8163 


0.1277 


14 


0.0910 


0.1820 


0.2730 


O.3640 


0.4550 


o.5459 


0.6369 


0. 7279 


0.8189 


0.1281 


15 


0.0913 


0.1826 


0.2738 


0.3651 


0.4564 


o.5477 


0.6389 


0.7302 


0.8215 


0.1285 


16 


0.0916 


0.1831 


0.2747 


0.3662 


o.4578 


0-5494 


0.6409 


o.7325 


0.8240 


0. 1289 


17 


c.0918 


0.1837 


0.2755 


O.3674 


0.4592 


0.55H 


0.6429 


0.7348 


0.8266 


0.1293 


18 


0.0921 


0.1843 


0.2764 


0.3685 


0.4607 


0.5528 


0.6449 


o.737i 


0.8292 


0.1297 


19 


0.0924 


0.1848 


0.2773 


0.3697 


0.4621 


o.5545 


0.6469 


o.7394 


0.8318 


0.1301 


20 


0.0927 


0.1854 


0.2781 


0.3708 


0.4635 


0.5562 


0.6489 


0.7416 


0.8343 


0.1305 


21 


0.0930 


0.1860 


0.2790 


O.3720 


0.4649 


o.5579 


0.6509 


o.7439 


0.8369 


0.1309 


22 


0.0933 


0.1865 


0.2798 


0.373I 


0.4664 


o.5596 


0.6529 


0. 7462 


0.8394 


0.1313 


23 


0.0936 


0.1871 


0.2807 


0.3742 


0.4678 


0.5614 


0.6549 


0.7485 


0.8420 


0.1317 


24 


0.0938 


0.1877 


0.2815 


0.3754 


0.4692 


0.5631 


0.6569 


0.7507 


0.8446 


0.132 1 


25 


0.0941 


0.1883 


0.2824 


0.3705 


0.4706 


0.5648 


0.6589 


0.7530 


0.8472 


0.1326 


26 


0.0944 


0.1888 


0.2833 


0.3777 


0.4721 


0.5665 


0.6609 


o.7553 


0.8498 


0.1330 


27 


0.0947 


0. 1894 


0.2841 


0.3788 


o.4735 


0.5682 


0.6629 


o.7576 


0.8523 


O.I334 


28 


0.0950 


0.1900 


0.2850 


0.3800 


o.4749 


0.5699 


0.6649 


o.7599 


0.8549 


0.1338 


29 


0.0953 


0.1905 


0.2858 


0.38 1 1 


0.4764 


0.5716 


0.6669 


0. 7622 


o.8574 


0.1342 


30 


0.0956 


0.1911 


0.2867 


O.3822 


o.4778 


C5734 


0.6689 


0.7645 


0.8600 


0.1346 


3i 


0.0958 


0.1917 


0.2875 


0.3834 


0.4792 


o.575i 


0.6709 


0. 7667 


0.8626 


0.1350 


32 


0.0961 


0.1923 


0.2884 


0.3845 


0.4806 


0.5768 


0.6729 


0.7690 


0.8652 


O.I354 


33 


0.0964 


0. 1928 


0.2892 


0.3856 


0.4820 


0.5785 


0.6749 


0.7713 


0.8677 


0.1358 


34 


0.0967 


0. 1934 


0.2901 


0.3868 


0.4835 


0.5802 


0.6769 


0.7736 


0.8703 


0. 1362 


35 


0.0970 


0. 1940 


0.2909 


0.3879 


0.4849 


0.5819 


0.6789 


o.7759 


0.8728 


0. 1366 


36 


0.0973 


0.1945 


0.2918 


0.3891 


0.4863 


0.5836 


0.6809 


0.7782 


0.8754 


0.1370 


37 


0.0976 


0.195 I 


0.2927 


0.3902 


0.4878 


0.5853 


0.6829 


0. 7804 


0.8780 


O.I374 


38 


0.0978 


0.1957 


0.2935 


0.3914 


0.4892 


0.5870 


0.6849 


0.7827 


0.8806 


0. 1378 


39 


0.0981 


0. 1962 


0.2944 


0.3925 


0.4906 


0.5887 


0.6869 


0.7850 


0.8831 


0.1382 


40 


0.0984 


c.1968 


0.2952 


O.3936 


0.4920 


0.5905 


0.6889 


0.7873 


0.8857 


0.1389 


41 


0.0987 


0.1974 


0.2961 


o.3948 


o.4935 


0.5921 


0.6908 


0.7895 


0.8882 


0. 1 390 


42 


0.0990 


0.1979 


0.2969 


0.3959 


0.4948 


o.5938 


0.6928 


0.7918 


0.8907 


O.I395 


43 


0.0993 


0.1985 


0.2978 


O.3970 


0.4963 


o.5956 


0.6948 


0.7941 


0.8933 


0.1399 


44 


0.0995 


0.1991 


0.2986 


0.3982 


0.4977 


o.5973 


0.6968 


0.7963 


0.8959 


0. 1403 


45 


0.0998 


0.1997 


0.2995 


o.3993 


0.4991 


0.5990 


0.6988 


0. 7986 


0.8985 


0. 1407 


46 


O.IOOI 


0.2002 


0.3003 


0.4004 


0.5006 


0.6007 


0. 7008 


0.8009 


0.9010 


0.1411 


47 


0. 1004 


0.2008 


0.3012 


0.4016 


0.5020 


0.6024 


0. 7028 


0.8032 


0.9036 


0.1415 


48 


0. 1007 


0.2014 


0.3020 


0.4027 


0.5034 


0.6041 


0. 7048 


0.8054 


0.9061 


0.1419 


49 


O.IOIO 


0.2019 


0.3029 


O.4039 


0.5049 


0.6058 


0.7068 


0.8078 


0.9087 


0.1423 


50 


0.1013 


0.2025 


0.3038 


O.4050 


0.5063 


0.6075 


0.7088 


0.8 ICO 


0.91 13 


0.1427 


5i 


0.1015 


0.2031 


0.3046 


0.4062 


0.5077 


0.6092 


0.7108 


0.8123 


0.9139 


0.1431 


52 


0.1018 


0.2036 


0.3055 


O.4073 


0.5091 


0.6109 


0.7127 


0.8146 


0.9164 


O.I435 


53 


0. 102 1 


0.2042 


0.3063 


O.4084 


0.5105 


0.6126 


0.7147 


0.8168 


0.9189 


0.1439 


54 


0. 1024 


0. 2048 


0.3072 


0.4096 


0.51 19 


0.6143 


0.7167 


0.8191 


0.9215 


O.I443 


55 


0. 1027 


0.2053 


0.3080 


0.4107 


o.5i34 


0.6160 


0.7187 


0.8214 


0.9240 


o.i447 


56 


0. 1030 


0.2059 


0.3089 


0.4118 


0.5148 


0.6177 


0. 7207 


0.8237 


0.9266 


0.145 1 


57 


0. 1032 


0.2065 


0.3097 


0.4130 


0.5162 


0.6194 


0.7227 


0.8259 


0.9292 


0. 1455 


58 


0.1035 


0.2071 


0.3106 


0.4141 


0.5176 


0.6212 


0.7247 


0.8282 


0.9318 


0.1459 


59 


0.1038 


0.2076 


0.31 14 


o.4!53 


0.5191 


0.6229 


0. 7267 


0.8305 


o.9343 


0.1463 


60 



100 


DISTANCES. 6° 


oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 


a 


0.9877 


1-9754 


2.9631 


3.95o8 


4.9384 


5.9261 


6.9138 


7.90I5 


8.8892 


1.3923 


OI 


0.9876 


1-9753 


2.9629 


3.9505 


4.9381 


5-9258 


6.9134 


7.9010 


8.8887 


1-3923 


02 


0.9876 


*-975* 


2.9627 


3-9503 


4-9378 


5.9254 


6.9130 


7.9005 


8.8881 


1.3922 


03 


0.9875 


1 -975o 


2.9625 


3-9500 


4-9375 


5-925o 


6.9125 


7.9000 


8.8875 


1.3922 


04 


0.9874 


1.9749 


2.9623 


3.9498 


4-9372 


5.9247 


6.9121 


7.8996 


8.8870 


1.3921 


05 


0.9874 


1.9748 


2.9621 


3-9495 


4.9369 


5-9243 


6.9117 


7.8991 


8.8864 


1.3921 


06 


0.9873 


1.9746 


2.9620 


3-9493 


4.9366 


5.9239 


6.9113 


7.8986 


8.8859 


1. 3921 


07 


0.9873 


1-9745 


2.9618 


3.9490 


4-9363 


5-9236 


6.9108 


7.8981 


8.8853 


1.3920 


08 


0.9S72 


1.9744 


2.9616 


3.9488 


4.9360 


5-9232 


6.9104 


7.8976 


8.8848 


1.3920 


09 


0.9871 


1-9743 


2.9614 


3.9486 


4-9357 


5.9228 


6.9100 


7.8971 


8.8842 


i-39*9 


10 


0.9871 


1-9742 


2.9612 


3-9483 


4-9354 


5.9225 


6.9095 


7.8966 


8.8837 


I-39I9 


II 


0.9870 


1.9740 


2.9610 


3.9481 


4.9351 


5.9221 


6.9091 


7.8961 


8.8831 


1.39*9 


12 


0.9870 


1-9739 


2.9609 


3-9478 


4.9348 


5.9217 


6.9087 


7.8956 


8.8826 


1-39*8 


13 


0.9869 


1-9738 


2.9607 


3-9476 


4-9344 


5-92I3 


6.9082 


7-895I 


8.8820 


1-39*8 


*4 


0.9868 


1-9737 


2.9605 


3-9473 


4-9341 


5.9210 


6.9078 


7.8946 


8.8814 


1-39*7 


15 


0.9868 


1-9735 


2.9603 


3-9471 


4-9338 


5.9206 


. 6.9073 


7.8941 


8.8809 


1-39*7 


16 


0.9867 


1-9734 


2.9601 


3.9468 


4-9335 


5.9202 


6.9069 


7.8936 


8.8803 


1-39*7 


*7 


0.9866 


1-9733 


2*9599 


3-9465 


4-9332 


5.9198 


6.9065 


7-8931 


8.8797 


1. 39*6 


18 


0.9866 


1.9732 


2-9597 


3-9463 


4.9329 


5-9*95 


6.9060 


7.8926 


8.8792 


1.3916 


*9 


0.9865 


1.9730 


2-9595 


3.9460 


4.9326 


5-9*9* 


6.9056 


7.8921 


8.8786 


1-39*5 


20 


0.9864 


1.9729 


2.9593 


3-9458 


4.9322 


5.9187 


6.9051 


7.8916 


8.8780 


1-39*5 


21 


0.9864 


1.9728 


2.9592 


3-9455 


4-93*9 


5-9183 


6.9047 


7.8911 


8.8775 


1.39*5 


22 


0.9863 


1.9726 


2.9590 


3-9453 


4.9316 


5-9*79 


6.9042 


7.8906 


8.8769 


1-39*4 


23 


0.9863 


1-9725 


2.9588 


3-9450 


4-9313 


5-9*75 


6.9038 


7.8900 


8.8763 


1. 39*4 


24 


0.9862 


1.9724 


2.9586 


3-9448 


4.9310 


5-9i7i 


6.9033 


7.8895 


8.8757 


1-39*3 


2 5 


0.9861 


1.9723 


2.9584 


3-9445 


4.9306 


5.9168 


6.9029 


7.8890 


8.8751 


1-39*3 


26 


0.9861 


1. 9721 


2.9582 


3-9442 


4.9303 


5.9164 


6.9024 


7.8885 


8.8745 


1. 39*3 


27 


0.9860 


1.9720 


2.9580 


3.9440 


4.9300 


5.9160 


6.9020 


7.8880 


8.8740 


1. 3912 


28 


0.9859 


1.9719 


2.9578 


3.9437 


4.9297 


5-9I56 


6.9015 


7.8875 


8.8734 


1.3912 


29 


0.9859 


I-97I7 


2.9576 


3-9435 


4.9293 


5-9*5 2 


6.901 1 


7.8869 


8.8728 


I-39** 


30 


0.9858 


1.9716 


2-9574 


3-9432 


4.9290 


5.9I48 


6.9006 


7.8S64 


8.8722 


1. 391 1 


31 


0.9857 


I-97I5 


2.9572 


3.9429 


4.9287 


5.9144 


6.9002 


7.8S59 


8.8716 


1.3910 


32 


0.9857 


I.97I3 


2.9570 


3-9427 


4.9284 


5.9140 


6.8997 


7.8854 


8.8710 


1-39*0 


33 


0.9856 


1.9712 


2.9568 


3-9424 


4.9280 


5.9136 


6.8992 


7.8848 


8.8704 


*• 39*o 


34 


0.9855 


1.9711 


2.9566 


3.9422 


4.9277 


5-9*32 


6.8988 


7.8843 


8.8698 


1.3909 


35 


0.9855 


1.9709 


2.9564 


3-9419 


4.9274 


5.9128 


6.S983 


7.8838 


8.8692 


1.3909 


36 


0.9854 


1.9708 


2.9562 


3.9416 


4.9270 


5-9 I2 4 


6.8978 


7.8832 


8.8687 


1.3908 


37 


0.9853 


1.9707 


2.9560 


3-94*4 


4.9267 


5.9120 


6.S974 


7.8827 


8.8681 


1.3908 


38 


0.9853 


I-9705 


2.9558 


3-94" 


4.9264 


5.9116 


6.8969 


7.8822 


8.8675 


1-3907 


39 


0.9852 


1.9704 


2.9556 


3.9408 


4.9260 


5.9112 


6.S964 


7.8S17 


8.8669 


1.3907 


40 


0.9851 


I-9703 


2-9554 


3.9406 


4.9257 


5.9108 


6.8960 


7.881 1 


S.S663 


1.3906 


4i 


0.9851 


1.9701 


2.9552 


3-9403 


4.9254 


5.9104 


6.8955 


7.SSC6 


8.S657 


1.3906 


42 


0.9850 


1.9700 


2.9550 


3.9400 


4.9250 


5.9100 


6.S950 


7.S800 


8.8650 


I.5905 


43 


0.9849 


1.9699 


2.9548 


3-939 s 


4.9247 


59096 


6.S946 


7.S795 


8.8644 


1-3905 


44 


0.9849 


1.9697 


2.9546 


3-9395 


4.9244 


5.9092 


6.S941 


7.8790 


S.8638 


1-3904 


45 


0.9848 


1.9696 


2.9544 


3-9392 


4.9240 


5.9088 


6.8936 


7.87S4 


S.S632 


I-3904 


46 


0.9847 


1.9695 


2.9542 


3-9389 


4-9237 


5-9084 


6.8931 


7.8779 


S.S626 


1.3903 


47 


0.9847 


1.9693 


2.9540 


3.9387 


4.9233 


5.90S0 


6.8927 


7.8773 


8.S620 


1.3903 


48 


0.9846 


1.9692 


2.9538 


3-93S4 


4.9230 


5.9076 


6.S922 


7.8768 


8.8614 


1.3902 


49 


0.9S45 


1.9691 


2.9536 


3-938i 


4.9227 


5.9072 


6.8917 


7.S762 


8.860S 


I-39C2 


50 


0.9845 


1.9689 


2-9534 


3-9379 


4.9223 


5.9068 


6.8912 


7.S757 


8.S602 


1.3901 


5i 


0.9S44 


1.968S 


2-9S32 


3.9376 


4.9220 


5.9064 


6.8908 


7.8752 


8.S595 


1.3901 


52 


0.9S43 


1.9686 


2.9530 


3-9373 


4.9216 


5.9059 


6.S903 


7.8746 


S.S5S9 


1.3900 


53 


0.9843 


1.9685 


2.9528 


3.9370 


4.9213 


5-9055 


6.S898 


7.S740 


S.85S3 


1.3900 


54 


0.9842 


1.9684 


2.9526 


3-9367 


4.9209 


5-905* 


6.S893 


7.8735 


S.S577 


I-3S99 


55 


0.9841 


1.9682 


2.9523 


3-9365 


4.9206 


5-9°47 


6.8888 


7.8729 


S.S^o 


I-3S99 


56 


0.9840 


1. 9681 


2.9521 


3.9362 


4.9202 


5-9043 


6.SS83 


7.S724 


S.S564 


1.3S9S 


57 


0.9S40 


1.9680 


2.9519 


3-9359 


4.9199 


5-9039 


6.8878 


7.8718 


S.S558 


1.389S 


58 


0.9839 


1.9678 


2.9517 


3-9356 


4-9*95 


5-9034 


6.8S74 


7.S713 


8.8552 


1.3897 


59 


0.9838 


1.9677 


2.9515 


3-9354 


4.9192 


5-9030 


6.SS69 


7.S707 


S.8545 


1.3S97 


60 


0.9838 


1-9675 


2.9513 


3-9351 


4.91SS 


5.9026 


6.8S64 


7.8702 


S.S539 


1.3896 



6° HEIGHTS. 101 


1 


3 


3 


4 


5 


6 


7 


8 


9 ; 


b 


00 


0.1038 


0.2076 


0.31 14 


o.4i53 


0.5191 


0.6229 


0.7267 


0.8305 


0.9343 


0.1463 


0. 1041 


0. 2082 


0.3123 


0.4164 


0.5205 


0.6246 


0.7287 


0.8327 


0.9368 


0.1467 


01 


0. 1044 


0.2088 


0.3131 


0.4I75 


0.5219 


0.6263 


0.7307 


0.8350 


0.9394 


0.1471 


02 


0. 1047 


0.2093 


0.3140 


0.4186 


0.5233 


0.6280 


0.7326 


0.8373 


0.9419 


0.1476 


03 i 


0. 1049 


0.2099 


0.3148 


0.4198 


0.5247 


0.6297 


0.7346 


0.8396 


0.9445 


0. 1480 


04 


0. 1052 


0.2105 


0.3157 


0.4209 


0.5262 


0.6314 


0.7366 


0.8418 


0.9471 


0. 1484 


05 


0.1055 


0.2IIO 


0.3165 


0.4220 


0.5276 


0.6331 


0.7386 


0.8441 


0.9496 


0. 1488 


06 


0.1058 


0.2116 


0.3174 


0.4232 


0.5290 


0.6348 


0. 7406 


0.8464 


0.9522 


0. 1492 


07 


0.1061 


0.2122 


0.3182 


0.4243 


0.5304 


0.6365 


0.7426 


0.8486 


0.9547 


0. 1496 


08 


0.1064 


0.2127 


0.3191 


0.4255 


0.5318 


0.6382 


0.7446 


0.8509 


0.9573 


0.1500 


09 


0. 1067 


0.2133 


0.3200 


0.4266 


0.5333 


0.6399 


0.7466 


0.8532 


0.9599 


0. 1504 


10 


0. 1069 


0.2139 


0.3208 


O.4277 


o.5347 


0.6416 


0.7485 


o.8554 


0.9624 


0.1508 


11 


0.1072 


0.2144 


0.3217 


0.4389 


0.5361 


0.6433 


0.7505 


0.8577 


0.9650 


0.1512 


12 


0.1075 


0.2150 


0.3225 


0.4300 


o.5375 


0.6450 


0.7525 


0.8600 


0.9675 


0.1516 


13 


0.1078 


0.2156 


0.3233 


0.4311 


0.5389 


0.6467 


o.7545 


0.8622 


0.9700 


0.1520 


14 


0.108 1 


0.2161 


0.3242 


0.4323 


c.5403 


0.6484 


0.7565 


0.8645 


0.9726 


0.1524 


15 


0.1084 


0.2167 


0.3251 


o.4334 


0.5418 


0.6501 


0.7585 


0.8668 


0.9752 


0.1528 


16 


0. 1086 


0.2173 


0.3259 


0.4346 


0-5432 


0.6518 


0.7605 


0.8691 


0.9778 


0.1532 


17 


0.1089 


0.2178 


0.3268 


o.4357 


0.5446 


0.6535 


0.7624 


0.8714 


0.9803 


0.1536 


18 


0.1092 


0.2184 


0.3276 


0.4368 


0.5460 


0.6552 


0.7644 


0.8736 


0.9828 


0.1540 


19 


0.1095 


0.2190 


0.3285 


0.4380 


0.5474 


0.6569 


0.7664 


0.8759 


0.9854 


0.1544 


20 


0.1098 


0.2195 


0.3293 


0.4391 


0.5488 


0.6586 


0. 7684 


0.8782 


0.9879 


0.1548 


21 


O.IIOI 


0.2201 


0.3302 


0.4402 


0.5503 


0.6603 


0.7704 


0.8804 


0.9905 


0.1552 


22 


0.1 103 


0.2207 


0.3310 


0.4414 


o.55i7 


0.6620 


0.7724 


0.8827 


0.9931 


0.1556 


23 


0.1 106 


0.2212 


0.3319 


0.4425 


o.553i 


0.6637 


o.7743 


0.8850 


0.9956 


0.1561 


24 


0.1 109 


0.2218 


0.3327 


0.4436 


0.5545 


0.6654 


0.7763 


0.8872 


0.9981 


0.1565 


25 


O. III2 


0.2224 


0.3336 


0.4448 


o.5559 


0.6671 


0.7783 


0.8889 


1.0007 


0.1569 


26 


O.III5 


0.2229 


0.3344 


o.4459 


o.5573 


0.6688 


0.7803 


C.8918 


1.0032 


O.I573 


27 


0,IIl8 


C.2235 


0.3353 


0.4470 


0.5588 


0.6705 


0.7823 


0.8940 


1x058 


O.I577 


28 


0.1 120 


0.2241 


0.3361 


0.4481 


0.5602 


0.6722 


0. 7842 


0.8963 


1.0083 


0.1581 


29 


0.1 123 


C.2246 


0.3370 


0.4493 


0.5616 


0.6739 


0. 7862 


0-8986 


1.0109 


0.1585 


30 


0.1 126 


0.2252 


0.3378 


0.4504 


0.5630 


0.6756 


0.7882 


0.9008 


1. 0134 


0.1589 


3i 


0.1 129 


0.2258 


0.3386 


o.45i5 


0.5644 


0.6773 


0. 7902 


0.9031 


1.0159 


0. 1593 


32 


0.1 132 


0.2263 


0.3395 


0.4527 


0.5659 


0.6790 


c. 7922 


0.9054 


1.0185 


O.I597 


33 


0.1 134 


0.2269 


0.3403 


0.4538 


0.5673 


0.6807 


0.7941 


0.9076 


1. 0210 


0.1601 


34 


0.1 137 


0.2275 


0.3412 


0.4549 


0.5687 


0.6824 


0. 7961 


0.9098 


1.0236 


0.1605 


35 


0.1 140 


0.2280 


0.3421 


0.4561 


0.5701 


0.6841 


0.8081 


0.9121 


1.0262 


0. 1609 


36 


0.1143 


0. 2286 


0.3429 


o.4572 


0.57I5 


C.6858 


0.8001 


0.9144 


1.0287 


0.1613 


37 


0.1 146 


0.2292 


0.3437 


0.4583 


0.5729 


0.6875 


0.8021 


0.9166 


1.0312 


0.1617 


38 


0. 1 149 


0.2297 


0.3446 


o.4594 


o.5743 


0.6892 


0.8040 


0.9189 


1-0337 


0.1621 


39 


0,1151 


0.2303 


0.3454 


0.4606 


o.5757 


0.6909 


0.8060 


0.9212 


1.0363 


0. 1625 


40 


0.1 154 


0.2309 


0.3463 


0.4617 


o.577i 


0.6926 


0.8080 


0.9234 


1.0389 


0.1629 


4 1 


0.1157 


0.2314 


Q-347 1 


0.4628 


0.5786 


0.6943 


0.8100 


0.9257 


1. 0414 


0.1633 


42 


0.1 160 


0.2320 


0.3480 


0.4640 


0.5800 


0.6960 


0.8119 


0.9279 


1.0439 


0.1637 


43 


0.1 163 


0.2326 


0.3488 


0.4651 


0.5814 


0.6977 


0.8139 


0.9302 


1.0465 


0.1641 


44 


0.1 166 


0.2331 


0.3497 


0.4662 


0.5828 


0.6994 


0.8159 


0.9325 


1.0490 


0.1645 


45 


0.1 168 


0.2337 


0.3505 


0.4674 


0.5842 


0.7010 


0.8179 


0.9347 


1. 0516 


0. 1650 


46 


0.1171 


0.2342 


0.3514 


0.4685 


0.5856 


0.7027 


0.8199 


0.9370 


1. 0541 


0.1654 


47 


0.1174 


0.2348 


0.3522 


0.4696 


0.5870 


0.7045 


0.8219 


o.9393 


1.0567 


0.1658 


48 


0.1177 


0.2354 


o.353i 


0.4708 


0.5884 


0. 7061 


0.8238 


0.9415 


1.0592 


0. 1662 


49 


0.1 180 


0.2359 


o.3539 


0.4719 


0.5899 


0. 7078 


0.8258 


0.9438 


1. 0617 


0.1666 


50 


0.1 183 


0.2365 


0.3548 


0.4730 


0.59I3 


0.7095 


0.8278 


0.9460 


1.0643 


0.1670 


5i 


0.1185 


0.2371 


o.3556 


0.4742 


0.5927 


0.7112 


0.8298 


0.9483 


1.0669 


0. 1674 


52 


0.1188 


0.2376 


0.3565 


o.4753 


0.5941 


0.7129 


0.8317 


0.9506 


1.0694 


0.1678 


53 


0.1 191 


0.2382 


o.3573 


0.4764 


o.5955 


0.7146 


0.8337 


0.9528 


1.0719 


0. 1682 


54 


0.1 194 


0.2388 


0.3581 


o.4775 


0.5969 


0.7163 


0.8357 


0.9550 


1.0744 


0. 1686 


55 


0.1 197 


0.2393 


0.3590 


0.4786 


0.5983 


0.7180 


0.8376 


o.9573 


1.0769 


0.1690 


56 


0.1 199 


0.2399 


0.3598 


0.4798 


0-5997 


0.7197 


0.8396 


0.9596 


1.0795 


0.1694 


57 


0.1202 


0.2405 


0.3607 


0.4809 


0.601 1 


0.7214 


0.8416 


0.9618 


1. 0821 


0.1698 


58 


0.1205 


0.2410 


0.3615 


0.4820 


0.6025 


0.7231 


0.8436 


0.9641' 


1 .0846 


0.1702 


59 


0.1208 


0.2416 


0.3624 


0.4832 


0.6040 


o.7247 


0.8455 


0.9663 


1. 0871 


0.1706 


60 



102 


DISTANCES. 7° 


oo 


1 


3 


3 


4 


5 


6 


7 


8 


9 


a 


0.9838 


1.9675 


2.9513 


3-9351 


4.9188 


5.9026 


6.8864 


7.8702 


8.8539 


1.3896 


OI 


0.9837 


1.9674 


2.9511 


3.9348 


4.9185 


5.9022 


6.8859 


7.8696 


8.8533 


1.3896 


02 


0.9836 


I.9673 


2.9509 


3-9345 


4.9181 


5.9018 


6.8854 


7.8690 


8.8526 


1-3895 


03 


0.9836 


1.9671 


2.9507 


3.9342 


4.9178 


5-90I3 


6.8849 


7.8684 


8.8520 


1.3895 


04 


0.9835 


1.9670 


2.9505 


3-9339 


4-9 x 74 


5.9009 


6.8844 


7.8679 


8.8514 


1.3894 


05 


0.9834 


1.9668 


2.9502 


3-9337 


4.9171 


5-9005 


6.8839 


7.8673 


8.8507 


L3894 


06 


o-9833 


1.9667 


2.9500 


3-9334 


4.9167 


5.9001 


6.8834 


7.8667 


8.8501 


1-3893 


07 


0.9833 


1.9665 


2.9498 


3-9331 


4.9164 


5.8996 


6.8829 


7.8662 


8.8494 


1.3893 


08 


0.9832 


1.9664 


2.9496 


3-9328 


4.9160 


5.8992 


6.8824 


7.8656 


8.8488 


1.3892 


09 


0.9831 


1.9663 


2-9494 


3-9325 


4.9156 


5.8988 


6.8819 


7.8650 


8.8482 


1.3892 


10 


0.9831 


1. 9661 


2.9492 


3-9322 


4-9 J 53 


5.8983 


6.8814 


7.8645 


8.8475 


1.3891 


II 


0.9830 


1.9660 


2.9490 


3.9319 


4.9149 


5.8979 


6.8809 


7.8639 


8.8469 


1.3891 


12 


0.9829 


1.9658 


2.9487 


3-93i6 


4.9146 


5-8975 


6.8804 


7.8633 


8.8462 


1.3890 


13 


0.9828 


i.9 6 57 


2.9485 


3.9314 


4.9142 


5.8970 


6.8799 


7.8627 


8.8456 


1.3890 


14 


0.9828 


I.9655 


2.9483 


3-93" 


4.9138 


5.8966 


6.8794 


7.8621 


8.8449 


1.3889 


15 


0.9827 


1.9654 


2.9481 


3-9308 


4-9J35 


5.8962 


6.8789 


7.8616 


8.8442 


1.3889 


16 


0.9826 


1.9652 


2.9479 


3-9305 


4-9 I 3* 


5-8957 


6.8783 


7.8610 


8.8436 


1.3888 


17 


0.9825 


1.9651 


2.9476 


3.9302 


4.9127 


5-8953 


6.8778 


7.8604 


8.8429 


1.3888 


18 


0.9825 


1.9650 


2-9474 


3.9299 


4.9124 


5-8949 


6.8773 


7.8598 


8.8423 


1.3887 


J 9 


0.9824 


1.9648 


2.9472 


3.9296 


4.9120 


5.8944 


6.8768 


7.8592 


8.8416 


1.3887 


20 


0.9823 


1.9647 


2.9470 


3-9293 


4.9117 


5.8940 


6.8763 


7-8586 


8.8410 


1.3886 


21 


0.9823 


I.9645 


2.9468 


3.9290 


4.9II3 


5-8935 


6.8758 


7.8580 


8.8403 


1.3886 


22 


0.9822 


1.9644 


2.9465 


3.9287 


4.9109 


5-893I 


6.8753 


7-8574 


8.8396 


1.3885 


23 


0.9821 


1.9642 


2.9463 


3.9284 


4.9105 


5.8926 


6.8747 


7.8569 


8.8390 


1.3885 


24 


0.9820 


1. 9641 


2.9461 


3.9281 


4.9102 


5.8922 


6.8742 


7.8563 


8.8383 


1.3884 


25 


0.9820 


1.9639 


2-9459 


3.9278 


4.9098 


5.8918 


6.8737 


7-8557 


8.8376 


1.3884 


26 


0.9819 


1.9638 


2-9457 


3-9275 


4.9094 


5-89I3 


6.8732 


7.8551 


8.8370 


1.3883 


27 


0.9818 


1.9636 


2-9454 


3.9272 


4.9091 


5.8909 


6.8727 


7-8545 


8.8363 


1.3883 


28 


0.9817 


I.9635 


2.9452 


3.9269 


4.9087 


5.8904 


6.8722 


7-8539 


8.8356 


1.3882 


29 


0.9817 


I.9633 


2.9450 


3.9266 


4.9083 


5.8900 


6.8716 


7.8533 


8.8349 


1.3S82 


30 


0.9816 


1.9632 


2.9448 


3-9263 


4.9079 


5.8895 


6.8711 


7-8527 


8.8343 


1.3S81 


3i 


0.9815 


1.9630 


2-9445 


3.9260 


4.9076 


5-8891 


6.8706 


7.8521 


8.8336 


1.3881 


32 


0.9814 


1.9629 


2-9443 


3.9257 


4.9072 


5.8886 


6.8700 


7.8515 


8.8329 


1.3880 


33 


0.9814 


1.9627 


2.9441 


3-9254 


4.9068 


5.8882 


6.8695 


7.8509 


8.8322 


1.3880 


34 


0.9813 


1.9626 


2.9438 


3-9251 


4.9064 


5.8877 


6.8690 


7-8503 


8.S315 


1.3879 


35 


0.9812 


1.9624 


2.9436 


3.9248 


4.9060 


5-8872 


6.8684 


7.8497 


8.S309 


1-3879 


36 


0.9811 


1.9623 


2-9434 


3.9245 


4-9057 


5.8S6S 


6.8679 


7.S490 


8.8302 


1.3S78 


37 


0.981 1 


1. 9621 


2.9432 


3.9242 


4-9053 


5.8863 


6.8674 


7.S4S4 


8.8295 


1.3S78 


38 


0.9810 


1.9620 


2.9429 


3.9239 


4.9049 


5-8859 


6.8669 


7.8478 


8.8288 | 1.3S77 


39 


0.9809 


1. 9618 


2.9427 


3.9236 


4-9045 


5-8854 


6.8663 


7.S472 


8.8281 i 


1.3S76 


40 


0.980S 


1. 9617 


2.9425 


3-9233 


4.9041 


5-8850 


6.8658 


7.8466 


8.8274 ! 


1.3876 


4i 


0.9807 


1.96*5 


2.9422 


3.9230 


4-9037 


5-8845 


6.S652 


7.S460 S.8267! 


1.3S75 


42 


0.9S07 


1.9613 


2.9420 


3.9227 


4-9034 


5.8840 


6.S647 


7.S454 1 8.S260 ! 


I-3S75 


43 


0.9806 


1. 9612 


2.941S 


3.9224 


4.9030 


5.8836 


6.S642 


7.844S j 8.S253 


I-3874 


44 


0.9805 


1. 9610 


2.9416 


3.9221 


4.9026 


5-8831 


6.S636 


7.8441 | S.S247 1.3S74 


45 


0.9804 


1.9609 


2.9413 


3.921S 


4.9022 


5.8826 


6.8631 


7.S435 1 S.S240 1.3S73 


46 


0.9804 


1.9607 


2.941 1 


3-9214 


4.901S 


5.8S22 


6.8625 


7.S429 S.S233 1.3S72 


47 


0.9803 


1.9606 


2.9409 


3-92H 


4.9014 


5-8Si 7 


6.S620 


7-8423 


8.8226 : I.3S72 


48 


0.9802 


1.9604 


2.9406 


3.9208 


4.9010 


5.8812 


6.S614 


7.S416 


8.S219 


1.3871 


49 


0.9801 


1.9603 


2.9404 


3-9205 


4.9006 


5.880S 


6.S609 


7.8410 


8.8212 


1.3871 


50 


0.9801 


1. 9601 


2.9402 


3.9202 


4.9003 


5-8803 


6.8604 


7.8404 


S.8205 


1.3S70 


5i 


0.9800 


1-9599 


2-9399 


3-9199 


4.S999 


5.8798 


6.8598 


7.S30S 8.8197 


1.3870 


52 


o.9799 


1.9598 


2-9397 


3.9196 


4.S995 


5.8794 


6.S592 


7.S391 S.S190 1 1.3S69 


53 


0.9798 


1.9596 


2-9394 


3-9I93 


4.8991 


5.S789 


6.8587 


7.S3S5 S.S1S3 1.3S6S 


54 


o.9797 


1-9595 


2.9392 


3.9189 


4.8987 


5.S7S4 


6.S5S1 


7.8379 : 8.8176 i 1.3S6S 


55 


o.9797 


1-9593 


2.9390 


3.9186 


4.8983 


5-8779 


6.S576 


7.S372 ; S.S169 ' 1.3S67 


56 


0.9796 


1.9592 


2.9387 


3-9183 


4.S979 


5-S775 


6.8570 


7.S366 , S.8162 1.3S66 


57 


o.9795 


1.9590 


2.9385 


3.9180 


4.8975 


5-8770 


6.S565 7.8^60 8.815s i.^S66 


58 


o.9794 


1.958S 


2.9383 


3-9*77 


4.S971 


5.S765 


6.S559 


7.8353 S.814S 1.3N55 


59 


o.9793 


1.9587 


2.9380 


3-9I73 


4.S967 


5.S760 


6.S554 


7.S347 S.81401 1.- 


60 


o.9793 


I.95S5 


2.9378 


3-9!70 


4.S963 


5-S755 


6.S54S 


7.8341 i 8.S133: 1.3S64 



r HEIGHTS. 




10:3 


1 


2 


3 


4 


5 


6 


7 


8 


9 


b 


00 


0.1208 


0.2416 


0.3624 


0.4832 


0.6040 


C.7247 


0.8455 


0.9663 


1. 0871 


0. 1 706 


0.1211 


0.2421 


0.3632 


0.4843 


0.6054 


0. 7264 


0.8475 


0.9686 


1.0896 


0.1710 


01 


0. 1214 


0.2427 


0.3641 


0.4854 


o.6c68 


0.7281 


0.8495 


0.9709 


1.0922 


0.1 714 


02 


0.1216 


0- 2 433 


0.3649 


0.4866 


0.6082 


0. 7298 


0.8515 


0-973 1 


1.0948 


0.1718 


03 


0.1219 


0.2438 


0.3658 


0.4877 


0.6196 


0.73I5 


0.8534 


o.9754 


!-0973 


0.1722 


04 


0.1222 


c.2444 


0.3666 


0.4888 


0.6110 


o.7332 


0.8554 


0.9776 


1.0998 


0.1726 


05 


0.1225 


0.2450 


0.3674 


0.4899 


0.6124 


0.7349 


0.8574 


0.9798 


1. 1023 


0.1730 


06 


0.1228 


0.2455 


0.3683 


0.4910 


0.6138 


0.7366 


0.8593 


0.9821 


1. 1048 


o.i734 


07 


0.1230 


0.2461 


0.3691 


0.4922 


0.6152 


0.7382 


0.8613 


0.9843 


1. 1074 


0.1738 


08 


0.1233 


0.2467 


0.3700 


0.4933 


0.6166 


0. 7400 


0.8633 


0.9866 


I.IIOO 


O.I743 


09 


0. 1236 


0.2472 


0.3708 


0.4944 


0.6180 


0.7417 


0.8653 


0.9889 


1.1125 


0.1747 


10 


0.1239 


0.2478 


o.37i7 


o.4956 


0.6194 


0.7433 


0.8672 


0.9911 


1.1150 


0.1751 


II 


0.1242 


0.2483 


0.3725 


0.4967 


0.6209 


o.745o 


0.8692 


o.9934 


I.II75 


O.I755 


12 


0.1245 


0.2489 


o.3734 


0.4978 


0.6223 


0.7467 


0.8712 


0.9956 


I.I20I 


O.I759 


13 


0.1247 


0.2495 


0.3742 


0.4989 


0.6237 


0.7484 


0.8731 


0.9978 


1. 1226 


0.1763 


14 


0.1250 


0.2500 


o.375o 


0.5000 


0.6251 


0.7501 


0.8751 


1. 0001 


I.I2SI 


0.1767 


15 


0.1253 


0.2506 


o.3759 


0.5012 


0.6265 


0.7518 


0.8771 


1.0024 


1. 1277 


0.1771 


16 


0.1256 


0.2512 


0.3767 


0.5023 


0.6279 


0.7535 


0.8791 


1.0046 


1. 1302 


O.I775 


17 


0.1259 


0.2517 


0.3776 


0.5034 


0.6293 


o.7552 


0.8810 


1.0069 


1. 1327 


0.1779 


l8 


0.1261 


0.2523 


0.3784 


0.5046 


0.6307 


0.7568 


0.8830 


1. 0091 


I- 1353 


0.1783 


J 9 


0. 1264 


0.2528 


0.3793 


0.5057 


0.6321 


0.7585 


0.8849 


1.0114 


1. 1378 


0.1787 


20 


0.1267 


0.2534 


0.3801 


0.5068 


0.6335 


0.7602 


0.8869 


1.0136 


I. I4O3 


0.1 791 


21 


0.1270 


0.2540 


0.3809 


0.5079 


0.6349 


0.7619 


0.8889 


1.0158 


1. 1428 


0.1795 


22 


0.1273 


0.2545 


0.3818 


0.5090 


0.6363 


0.7636 


0.8908 


1.0181 


I- 1453 


0.1799 


23 


0.1275 


0.2551 


0.3826 


0.5102 


0.6377 


0.7652 


0.8928 


1.0203 


I. 1479 


0.1803 


24 


0.1278 


c.2556 


0.3835 


0.5H3 


0.6391 


0. 7669 


0.8947 


1.0226 


I. I5O4 


0.1807 


25 


0.1281 


0.2562 


0.3843 


0.5124 


0.6405 


0.7680 


0.8967 


1.0248 


I. I529 


0.1811 


26 


0.1284 


0.2568 


0.3852 


0.5136 


0.6419 


c.7703 


0.8987 


1. 0271 


I- 1555 


0.1815 


27 


0.1287 


o.2573 


0.3860 


o.5i47 


0.6433 


0.7720 


0.9007 


1.0294 


I. I580 


0.1819 


28 


0.1289 


0.2579 


0.3868 


0.5158 


0.6447 


o.7737 


0.9026 


1. 0316 


1. 1605 


0.1823 


29 


0.1292 


0.2585 


0.3877 


0.5169 


0.6461 


o.7754 


0.9046 


1.0338 


I.163I 


0.1827 


30 


0. 1295 


0.2590 


0.3885 


0.5180 


0.6475 


0.7771 


0.9066 


1. 0361 


1. 1656 


0.1831 


3i 


0.1298 


0.2596 


0.3894 


0.5192 


0.6489 


0.7787 


0.9085 


1-0383 


I.l68l 


0.1835 


32 


0.1301 


0.2601 


0.3902 


0.5203 


0.6503 


0. 7804 


0.9105 


1.0406 


1. 1 706 


0.1839 


33 


0.1303 


0.2607 


0.3910 


0.5214 


0.6517 


0.7821 


0.9124 


1.0428 


1. 1 731 


0.1843 


34 


0.1306 


0.2613 


0.3919 


0.5225 


0.6532 


0.7838 


0.9144 


1.0450 


I-I757 


0.1847 


35 


0.1309 


0.2618 


0.3927 


0.5236 


0.6546 


0.7855 


0.9164 


1 -04 73 


1. 1782 


0.1852 


36 


0.1312 


0.2624 


0.3936 


0.5248 


0.6560 


0.7871 


0.9183 


1.0495 


1. 1807 


0.1856 


37 


0.1315 


0.2629 


0.3944 


0-5259 


0.6574 


0.7888 


0.9203 


1.0518 


1. 1832 


0. i860 


38 


0.1318 


0.2635 


0-3953 


0.5270 


0.6588 


0.7905 


0.9223 


1.0540 


1. 1858 


0.1864 


39 


0. 1320 


0.2641 


0.3961 


0.5281 


0.6602 


0. 7922 


0.9242 


1.0562 


1.1883 


0.1868 


40 


0.1323 


0. 2646 


0.3969 


0.5292 


0.6616 


0.7939 


0.9262 


1.0585 


1. 1908 


0.1872 


4i 


0. 1326 


0.2652 


0.3978 


0.5304 


0.6630 


o.7955 


0.9281 


1.0607 


1-^933 


0. 1876 


42 


0. 1329 


0.2657 


0.3986 


0.53I5 


0.6644 


0.7972 


0.9301 


1.0630 


1. 1958 


0.1880 


43 


0.1332 


0. 2663 


0.3995 


0.5326 


0.6658 


0.7989 


0.9321 


1.0652 


1. 1984 


0.1884 


44 


o-i334 


0.2669 


0.4003 


o.5337 


0.6672 


0.8006 


0.9340 


1.0674 


1.2009 


0.1888 


45 


0.1337 


0.2674 


0. 401 1 


o.5348 


0.6686 


0.8023 


0.9360 


1.0697 


1.2034 


0. 1892 


46 


0. 1340 


0.2680 


0.4020 


0.5360 


0.6700 


0.8039 


o.9379 


1.0719 


1.2059 


0.1896 


47 


0.1343 


0.2685 


0.4028 


o.537i 


0.6714 


0.8056 


o.9399 


1.0742 


1.2084 


0.1900 


48 


0.1346 


0.2691 


0.4037 


0.5382 


0.6728 


0.8073 


0.9419 


1.0764 


1.2110 


0.1904 


49 


0.1348 


0.2697 


0.4045 


05393 


0.6742 


0.8090 


0.9438 


1.0786 


1-2135 


0.1908 


50 


0.1351 


0.2702 


0.4053 


0.5404 


0.6756 


0.8107 


0.9458 


1.0809 


1. 2160 


0.1912 


5i 


0.1354 


0.2708 


0.4062 


0.5416 


0.6770 


0.8123 


0.9477 


1.0831 


1.2185 


0.1916 


52 


0.1357 


0.2713 


0.4070 


0.5427 


0.6783 


0.8140 


0.9497 


1.0854 


1. 2210 


0.1920 


53 


0.1359 


0.2719 


0.4078 


o.5438 


0.6797 


0.8157 


0.9516 


1.0876 


1.2235 


0.1924 


54 


0.1362 


0.2725 


0.4087 


0.5449 


0.6811 


O.8174 


o.9536 


1.0898 


1. 2261 


0. 1928 


55 


0.1365 


0.2730 


0.4095 


0.5460 


0.6825 


0^8191 


0.9556 


1. 0921 


1.2286 


0.1932 


56 


0.1368 


0.2736 


0.4104 


0.5472 


0.6839 


0.8207 


o.9575 


1.0943 


1.2311 


0.1936 


57 


0.1371 


0.2741 


0.4112 


0.5483 


0.6853 


0.8224 


o.9595 


1.0966 


1.2336 


0.1940 


58 


0.1374 


0.2747 


0.4121 


o.5494 


0.6867 


0.8241 


0.9615 


1.0988 


1.2362 


0.1944 


59 


0.1376 


0.2753 


0.4129 


0.5505 


0.6881 


0.8258 


0.9634 


I.IOIO 


1.2387 


0.1948 


60 



104 


DISTANCES. 


8= 


/ 
oo 


1 


3 


3 


4 5 


6 


7 


8 9 


a 


0-9793 


1.958s 


2.9378 


3.9170 4.8963 


5.8755 


6.8548 


7.8341 8.8133 


1.3864 


OI 


0.9792 


1.9584 


2-9375 


3.9167 4.8959 


5.8751 


6.8542 


7.S334 8.8126 


1.3863 


02 


0.9791 


1.9582 


2-9373 


3.9164 4.8955 


5.8746 


6.8537 


7.832S 8.8119 


1.3S62 


03 


0.9790 


1.95S0 


2.9370 


3.9161 4.S951 


5.8741 


6.8531 


, 7.8321 8.8111 


1.3862 


04 


0.9789 


1-9579 


2.9368 


3.9157 4.8947 


5-8736 


6.8525 


: 7.8315 8.S104 


1.3861 


05 


0.97S9 


1-9577 


2.9366 


3.9154 4.S943 


5.8731 


6.8520 


; 7.8308 8.8097 


1.3861 


06 


0.978s 


1-9575 


2.9363 


3.9I5I 4-8939 


5-8726 


6.8514 


7.8302 8.S090 


1.3S60 


07 


0.9787 


1-9574 


2.9361 


3.9148 4.8935 


5.8722 


6.S508 


7.8295 8.80S2 


1.3S60 


oS 


0.97S6 


1-9572 


2.935S 


3.9144 4.8931 


5-87I7 


6.8503 


7.8289 8.S075 


I.3S59 


09 


0.97S5 


I-957I 


2.9350 


3-9I4I 4-8927 


5.8712 


6.8497 


7.8282 8.806S 


1.3859 


10 


c.9785 


1-9569 


2-9354 


3-9I38 4-8923 


5-8707 


6.8492 


7.8276 t 8.8061 


1.3S58 


11 


0.9784 


I-9567 


2-9351 


3.9135 4.8918 


5-8702 


6.8486 


7.8269 8.8053 


1.3858 


12 


o-97S3 


1.9566 


2-9349 


3-9I3I 4.8914 


5.S697 


6.8480 


7.8263 8.8046 


1.3857 


13 


0.9782 


I-9564 


2.9346 


3.912S 4.S910 


5.8692 


6.8474 


7.8256 8.8038 


1.3856 


14 


0.9781 


1.9562 


2-9344 


3.9125 4.8936 


5.8687 


6.8468 


7.8250 : 8.8031 


1.3856 


15 


0.9780 


1. 9561 


2.9341 


3.9122 4.8902 


5.8682 


6.8463 


7-8243 


S.8023 


L3S55 


16 


0.9780 


1-9559 


2-9339 


3.9118 4.8898 


5.8677 


6.8457 


7.8236 


8.8016 


1.3854 


17 


0.9779 


1-9557 


2.9336 


3.9115 4.S894 


5-8672 


6.8451 


7.8230 


8.8009 


1.3S54 


iS 


0.9778 


I.9556 


2-9334 


3.9112 4.8890 


5.S667 


6.8445 


7.8223 


8.S001 


1.3853 


19 


0-9777 


1-9554 


2.9331 


3.910S 4.S8S5 


5.8662 


6.8440 


7.8217 


8.7994 


i-5>-5 


23 


0.9773 


1-9553 


2.9329 


3.9105 4.8881 


5-8657 


6.8434 


7.8210 


8.7986 


1.3852 


21 


o.9775 


1-9551 


2.9326 


3.9102 4.8877 


3-8652 


6.S42S 


7.8203 


8.7979 


1.3832 


22 


o.9775 


1-9549 


2.9324 


3.909S 4.8873 


5-8647 


6.8422 


7.8197 


8.7971 


1.3851 


23 


0-9774 


1-9547 


2.9321 


3.9095 4.8869 


5-8642 


6.8416 


7.8190 


S.7964 


1.3850 


24 


o.9773 


I-9546 


2.9319 


3.9392 4.SS64 


5.8637 


6.8410 


7.8183 


8.7956 


1.3S50 


25 


0.9772 


1-9544 


2.9316 


3.908S 4.SS60 


5-8632 


6.8404 


7.8176 


S.794S 


I.3S49 


26 


0.9771 


1-9542 


2.9314 


3.9085 4.8856 


5-8627 


6.S398 


7. Si 70 


8.7941 


1.3849 


27 


0.9770 


I-954I 


2.9311 


3.90S1 4.8S52 


5.S622 


6.S393 


7.8163 


S-7933 


1.3848 


28 


0.9770 


1-9539 


2.9309 


3.937S 4.8848 


5.8617 


6.S387 


7.8156 


S.7926 


I-3847 


2 9 


0.9769 


1-9537 


2.9306 


3.9075 4.8543 


5.S612 


6.83S1 


7.8149 


S.791S 


1.3847 


30 


0.9768 


I-953 5 


2.9304 


3.9371 4.S839 


3.8607 


6.S375 


7-SI43 


S.7911 


1.3S46 


31 


0.9767 


1-9534 


2.9301 


3.9068 4.8835 


5.S602 


6.8369 


7.8136 


8.7903 


1.3846 


32 


0.9766 


1-9532 


2.929S 


3.9364 4.8831 


5-8597 


6.8333 


7.S129 


S.7895 


1.3845 


33 


0.9765 


1 -9531 


2.9296 


3.9361 4.S826 


5.8592 


6.S357 


7.S122 


S.7S87 


I.3844 


34 


0.9764 


1-9529 


2.9293 


3.905S 4.8822 


0.85S6 


6.8351 


7.S115 


S.7SS0 


1.3844 


35 


0.9764 


1-9527 


2.9291 


3.9354 4.8818 


5-S5Si 


6.S345 


7.S108 


S.7872 


L3S43 


36 


0.9763 


I.9525 


2.92S8 


3.9051 4-SSi3 


5.8576 


6.S339 


7.8102 


S.7S64 


I.3S43 


37 


0.9762 


1-9524 


2.9285 


3.9047 4.8S09 


5.8571 


6.S333 


7-8095 


8.7S56 


1.3842 


38 


0.9761 


1.9522 


2.92S3 


3.9044 4.8S05 


5.8566 


6.S327 


7.8088 


S.7849 


1.3S41 


39 


0.9763 


1.9520 


2.9280 


3.9040 4.8801 


5.S561 


6.S321 


7.S0S1 


S.7S41 


1.3S41 


40 


0-9759 


I.95I9 


2.927S 


3.9037 4.S796 


5-S556 


6.S315 


7.S074 


8.7833 


1.3S40 


4i 


o.9758 


I.95I7 


2.9275 


3.9034 4-8792 


5-S55° 


6.S309 


7.8067 


8.7826 


1.3840 


+2 


0.9758 


I-95I5 


2.9273 


3.9030 4.S7SS 


5.S545 


6.S303 


7.S060 


S.781S 


I.3S39 


43 


o.9757 


I-95I3 


2.9270 


3.9027 4.87S3 


5-8540 


6.S296 


7-8053 


S.7810 


1.3838 


44 


0.9756 


I-95I2 


2.9267 


3.9023 4.8779 


5.8535 


6.8290 


7.S046 




1.3S3S 


45 


o.9755 


1. 9510 


2.9265 


3.9020 4.8774 


5-S529 


6.S2S4 


7.S039 


S-7794 


, #37 


46 


o.9754 


1.9508 


2.9262 


3.9016 4.S770 


5.S524 


6.S27S 


7.8-3- 


8.77S6 


I-3837 


47 


o.9753 


1.9506 


2.9259 


3.9313 4.S766 


3.S519 


6.S272 


7.8025 


S.777S 


1.3S36 


48 


o.9752 


1-9505 


2.9257 


3.9009 4.S761 


5-S5I4 


6.S266 


7.S01S 


S.7770 


I.3S35 


49 


o.975i 


I-9503 


2.9254 


3.9006 4.S757 


5.S50S 


6.S260 


7. Son 


S.7763 


L3S35 


5C 


o.975i 


1.9501 


2.9252 


3.9002 4.8 753 


5-S503 


6.S254 


7.8004 


S-7755 




5i 


! 0.9750 


1.9499 


2.9249 


3.S999 4.5748 


5.S498 


6.S247 


7-7997 


8.7747 




52 


o.9749 


1-9497 


2.9246 


3.S995 4.S744 


5.S492 


6.S241 


7-7990 


8. 7739 


I.3S33 


53 


C.974S 


1.9496 


2.9244 


3-8991 4-S739 


5-8487 


6.8235 


7-7983 


S.773I 


1.3S32 


54 


o.9747 


1-9494 


2.9241 


3.S9SS 4.S735 


5.S4S2 


6.S229 


7.7976 


S.7723 


13S32 


55 


0.9746 


1.9492 


2.923S 


3.S9S4 4.S730 


5.S476 


6.S222 


7-7969 


8.7715 




56 


o.9745 


1.9490 


2.9236 


3.S9S1 4.S726 


5.S47I 


6.8216 


7.7961 






57 


0.9744 


1.94S9 


2.9233 


3-8977 4-S72I 


5.8466 


6.S210 


7- 7954 


S.7699 




5S 


0-9743 


1.94S7 


2.9230 


3.S974 4.S717 


5-8460 


68204 


7-7947 


S.7691 




59 


o.9743 


I-94S5 


2.922S 


3.S970 4.S713 


5.8455 


6.S19S 


7- 7940 




1-3829 


60 


0.9742 


I.94S3 


2.9225 


3.S966 4.S708 


5-8450 


6.8191 


7-7933 


S.767- 


: j8a8 



8° HEIGHTS. 




105 


1 


2 


3 

0.4129 


4 

0.5505 


5 


6 


7 


8 


9 


b 


00 


0.1376 


0.2753 


0.6881 


0.8258 


0.9634 


I.IOIO 


1.2387 


0.1948 


0. 1380 


0.2758 


o-4 x 37 


0.5516 


0.6895 


0.8275 


0.9654 


1. 1033 


1. 2412 


0. 1952 


01 


c.1382 


0.2764 


0.4145 


0.5527 


0.6909 


c.8291 


0.9673 


I- 1055 


1-2437 


0.1956 


02 


0.1385 


0.2769 


0.4154 


0.5538 


0.6923 


0.8308 


0.9692 


1. 1077 


1.2462 


0. i960 


03 


0.1387 


0.2775 


0.4162 


0.5550 


0.6937 


0.8324 


0.9712 


1.1099 


T.2487 


0.1965 


°4 


0.1390 


0.2780 


0.4171 


0.5561 


0.6951 


0.8341 


0.9731 


1.1122 


1. 2512 


0. 1969 


05 


o.i393 


0.2786 


0.4179 


0.5572 


0.6965 


0.8358 


o.975i 


1. 1144 


!-2537 


O.I973 


06 


0.1390 


0.2792 


0.4187 


0.5583 


0.6979 


0.8375 


0.9771 


1.1166 


1.2562 


0.1977 


07 


0.1399 


0.2797 


0.4196 


o.5594 


0.6993 


0.8392 


0.9790 


1.1189 


1.2587 


0.1981 


08 


0. 1401 


0.2803 


0.4204 


0.5606 


0. 7007 


0.8408 


0.9810 


1.1211 


1. 2613 


0. 1985 


°9 


0.1404 


O.2808 


0.4213 


0.5617 


0.7021 


0.8425 


0.9829 


1. 1234 


1.2638 


0.1989 


10 


0. 1407 


0.2814 


0.4221 


0.5628 


0.7035 


0.8441 


0.9849 


1. 1256 


1.2663 


O.I993 


11 


0.1410 


0.2819 


0.4229 


0.5639 


0.7049 


0.8458 


0.9868 


1. 1278 


1.2688 


0.1597 


12 


0.1413 


0.2825 


0.4238 


0.5650 


0.7063 


0.8475 


0.9888 


1. 1300 


1. 2713 


0.2001 


13 


0.1415 


O.2831 


0.4246 


0.5661 


0.7077 


0.8492 


0.9907 


1. 1322 


1.2738 


0.2005 


H 


0.1418 


0.2836 


0.4254 


0.5672 


0. 7091 


0.8509 


0.9927 


1. 1345 


1.2763 


0.2c 09 


15 


0.142 1 


O.2842 


0.4263 


0.5684 


0.7104 


0.8525 


0.9946 


1. 1367 


1.2788 


0.2013 


16 


0. 1424 


0.2847 


0.4271 


0.5695 


0.7118 


0.8542 


0.9966 


1. 1390 


1. 2813 


0.2017 


17 


0.1426 


0.2853 


0.4279 


0.5706 


0.7132 


0.8558 


c.9985 


1.1412 


1.2838 


0.2021 


18 


0. 1429 


0.2858 


0.4288 


0.57I7 


0.7146 


0.8575 


1.0005 


1. H34 


1.2863 


0.2025 


J 9 


0.1432 


0.2864 


c.4296 


0.5728 


0.7160 


0.8592 


1.0024 


1. 1456 


1.2888 


0.2029 


20 


o.i435 


0.2870 


0.4304 


o.5739 


0.7174 


C.86C9 


1.0044 


1. 1478 


1.2913 


0.2033 


21 


0. 1438 


O.2875 


o-43 x 3 


0.5750 


0.7188 


0.8626 


1x063 


1.1501 


1.2938 


0.2037 


22 


0.1440 


0.2881 


0.4321 


0.5762 


0. 7202 


0.8642 


1.0083 


1. 1523 


1.2963 


0.2041 


23 


0.1443 


0.2886 


0.4329 


0-5773 


0.7216 


0.8659 


1. 0102 


1. 1545 


1.2988 


0.2045 


24 


0. 1446 


0.2892 


o-433 8 


0.5784 


0.7230 


0.8675 


1.0121 


1. 1567 


1. 3013 


0.2049 


25 


0.1449 


0.2897 


0.4346 


0.5795 


0.7243 


0.8692 


1.0141 


1. 1590 


1.3038 


0.2053 


26 


0.1451 


O.2903 


o.4354 


0.5806 


0.7257 


0,8709 


1.0160 


1.1612 


1-3063 


0.2057 


27 


0.1454 


0.2909 


o.43°3 


0.5817 


0.7271 


0.8726 


1. 0180 


1. 1634 


1.3088 


0.2061 


28 


o.i457 


O.2914 


o.437i 


0.5828 


0.7285 


0.8742 


1. 0199 


1. 1656 


1-3113 


0.2065 


29 


0. 1460 


0.2920 


o.4379 


0.5839 


0. 7299 


c.8759 


1. 02 19 


1. 1678 


1-3138 


0.2069 


30 


0. 1463 


O.2925 


0.4388 


0.5850 


0.73I3 


0.8776 


1.0238 


T.I 7OI 


i.3 J 63 


0.2073 


3i 


0. 1465 


0.2931 


0.4396 


0.5862 


0.7327 


0.8792 


1.0258 


1. 1723 


1.31S8 


0.2077 


32 


0. 1468 


O.2936 


0.4404 


c.5873 


o.734i 


0.8809 


1.0277 


1-1745 


L3213 


0.2081 


33 


0.1471 


0.2942 


0.4413 


0.5884 


o.7355 


0.8825 


1.0296 


1. 1767 


1.3238 


0.2085 


34 


0.1474 


0.2947 


0.4421 


0.5895 


0.7368 


c.8842 


1. 0316 


1. 1790 


1.3263 


0.2089 


35 


0.1476 


0.2953 


0.4429 


0.5906 


0.7382 


0.8859 


I-0335 


1.1812 


1.3288 


0.2093 


36 


0.1479 


O.2958 


0.4438 


0.59I7 


0.7396 


0.8875 


i.o355 


1. 1834 


I.33I3 


0.2097 


37 


0.1482 


0.2964 


0.4446 


0.5928 


0.7410 


0.8892 


1.0374 


1. 1856 


1.3338 


0.2101 


38 


0.1485 


O.2970 


0-4454 


o.5939 


0.7424 


0.8909 


1.0394 


1. 1878 


1-3363 


0.2105 


39 


0.1488 


0.2975 


0.4463 


0.5950 


0.7438 


0.8926 


1.0413 


1.1901 


1.3388 


0.2110 


40 


0. 1490 


O.2981 


0.4471 


0.5961 


o.7452 


0.8942 


1.0432 


1. 1923 


I-34I3 


0.2114 


4i 


0.1493 


O.2986 


o.4479 


o.5972 


0.7466 


0.8959 


1.0452 


1. 1945 


1.3438 


0.2118 


42 


0. 1496 


0.2992 


0.4488 


0.5984 


o.7479 


0.8975 


1.0471 


1. 1967 


1.3463 


0.2122 


43 


0.1499 


O.2997 


0.4496 


o.5995 


o.7493 


0.8992 


1. 0491 


1. 1989 


1.3488 


0.2126 


44 


0.1 501 


0.3003 


0.4504 


0.6006 


0.7507 


0.9008 


1. 0510 


1.20PI 


I-35I3 


0.2130 


45 


0. 1504 


O.3008 


o.45i3 


0.6017 


0.7521 


0.9025 


1.0529 


1.2634 


1.3538 


0.2134 


46 


0. 1507 


O.3014 


0.4521 


0.6028 


o.7535 


0.9042 


1 -0549 


1.2056 


1-3563 


0.2138 


47 


0.1510 


0.3019 


0.4529 


0.6039 


o.7549 


0.9058 


1.0568 


1.2078 


1.3588 


0.2142 


48 


0.1513 


0.3025 


o.4538 


0.6050 


0.7563 


0.9075 


1.0588 


1.21CO 


1.3613 


0.2146 


49 


0.1515 


0.3031 


0.4546 


0.6061 


0.7576 


0.9092 


1.0607 


1. 2122 


1.3638 


0.2150 


50 


0.1518 


0.3036 


o.4554 


0.6072 


0.7590 


0.9108 


1.0626 


1. 2144 


1.3662 


0.2154 


5i 


0.1521 


O.3042 


c.4562 


0.6083 


0. 7604 


C.9125 


1.0646 


1. 2166 


1.3687 


0.2158 


52 


0.1524 


0.3047 


Q.457 1 


0.6094 


0.7618 


c.9142 


1.0665 


1. 2189 


1. 3712 


0.2162 


53 


0.1526 


0.3053 


o.4579 


0.6105 


0.7632 


0.9158 


1.0684 


1. 2211 


1-3737 


0.2166 


54 


0.1529 


0.3058 


0.4587 


0.6116 


0. 7646 


0-9*75 


1.0704 


1.2233 


1.3762 


0.2170 


55 


0.1532 


0.3064 


0.4596 


0.6128 


0. 7660 


0.9191 


1.0723 


1.2255 


1.3787 


0.2174 


56 


o.i535 


O.3069 


0.4604 


0.6139 


0.7673 


0.9208 


1.0742 


1.2277 


1.3812 


0.2178 


57 


0.1537 


0.3075 


0.4612 


0.6150 


0.7687 


0.9224 


1.0762 


1.2299 


1.3837 | 


0.2182 


58 


0.1540 


0.3080 


0.4621 


0.6161 


0.7701 


0.9241 


1. 0781 


1. 2321 


1.3862 ! 


0.2186 


59 


0.1543 


0.3086 


0.4629 


0.6172 


o.77i5 


0.9257 


r.o8co 


1-2343 


1.3886 | 


0.2190 


60 



106 DISTANCES. 


9° 


/ 

oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 


a 


0.9742 


I-9483 


2.9225 


3.8966 


4.8708 


5-8450 


6.8191 


7-7933 


8.7675 


1.3828 


OI 


0.9741 


1. 9481 


2.9222 


3.8963 


4.8704 


5.8444 


6.8185 


7.7926 


8.7666 


1.3827 


02 


0.9740 


1.9480 


2.9219 


3.8959 


4.8699 


5-8439 


6.8179 


7.7918 


8.7658 


1.3826 


03 


0.9739 


1.9478 


2.9217 


3.8956 


4.8695 


5-8433 


6.8172 


7. 791 1 


8.7650 


1.3826 


04 


0.9738 


1.9476 


2.9214 


3.8952 


4.8690 


5.8428 


6.8166 


7.7904 


8.7642 


1.3825 


OS 


0.9737 


1.9474 


2.9211 


3.8948 


4.8686 


5-8423 


6.8160 


7.7897 


8.7634 


1-3825 


06 


0.9736 


1.9472 


2.9209 


3.8945 


4.8681 


5-84I7 


6.8153 


7.7890 


8.7626 


1.3824 


07 


o.9735 


I-947I 


2.9206 


3.8941 


4.8676 


5.8412 


6.8,147 


7.7882 


8.7618 


1.3824 


08 


o.9734 


1.9469 


2.9203 


3-8938 


4.8672 


5.8406 


6.8141 


7.7S75 


8.7609 


1.3823 


09 


o.9733 


1.9467 


2.9200 


3-8934 


4.8667 


5.8401 


6.8134 


7.7868 


8.7601 


1.3822 


10 


o.9733 


1.9465 


2.9198 


3-8930 


4.8663 


5.8395 


6.8128 


7.7861 


8-7593 


1.3821 


11 


0.9732 


L9463 


2.9195 


3.8927 


4.8658 


5.8390 


6.8122 


7.7853 


8.7585 


1.3821 


12 


0.9731 


1. 9461 


2.9192 


3-8923 


4.8654 


5.8384 


6.8115 


7.7846 


8-7577 


1.3820 


J 3 


0.9730 


1.9460 


2.9189 


3.8919 


4. 8649 


5.8379 


6.8109 


7.7838 


8.7568 


1.3819 


14 


0.9729 


1.9458 


2.9187 


3.8916 


4.8644 


5.8373 


6.8102 


7.7831 


8.7560 


1.3819 


15 


0.9728 


1.9456 


2.9184 


3.8912 


4.8640 


5.8368 


6.8096 


7.7824 


8.7552 


1.3818 


16 


0.9727 


1-9454 


2.9181 


3.8908 


4-8635 


5.8362 


6.8089 


7.7816 


8-7543 


1.3818 


17 


0.9726 


1.9452 


2.9178 


3.8904 


4.8631 


5.8357 


6.8083 


7.7809 


8-7535 


1.3817 


18 


0.9725 


1.9450 


2.9176 


3.8901 


4.8626 


5.8351 


6.8076 


7.7802 


8.7527 


1.3816 


19 


0.9724 


1.9449 


2.9 J 73 


3.8897 


4.8621 


5.8346 


6.8070 


7-7794 


8.7518 


1.3816 


20 


0.9723 


1-9447 


2.9170 


3-8893 


4.8617 


5-8340 


6.8063 


7.7787 


8.7510 


i-38i5 


21 


0.9722 


1-9445 


2.9167 


3.8890 


4.8612 


5.8334 


6.8057 


7.7779 


8.7502 


1-3814 


22 


0.9721 


1-9443 


2.9164 


3.8886 


4.8607 


5-8329 


6.8050 


7.7772 


8-7493 


1.3814 


23 


0.9721 


1.9441 


2.9162 


3.8882 


4.8603 


5.8323 


6.8044 


7-7764 


8.7485 


1-3813 


24 


0.9720 


1-9439 


2.9159 


3.8878 


4.8598 


5.8318 


6.8037 


7-7757 


8.7476 


1.3S13 


25 


0.9719 


1-9437 


2.9156 


3.8875 


4.8593 


5.8312 


6.8031 


7-7749 


8.7468 


1.3812 


26 


0.9718 


1-9435 


2.9153 


3.8871 


4.8589 


5.8306 


6.8024 


7-7742 


8.7460 


1.3811 


27 


0.9717 


1-9434 


2.9150 


3.8867 


4.8584 


5-8301 


6.8018 


7-7734 


8-7451 


1.3811 


28 


0.9716 


1-9432 


2.9148 


3.8863 


4-8579 


5-8295 


6.8011 


7-7727 


8-7443 


1. 3810 


29 


o.97 J 5 


1.9430 


2.9145 


3.8860 


4.8575 


5.8290 


6.8004 


7-77I9 


8-7434 


1.3S10 


30 


0.9714 


1.9428 


2.9142 


3.8856 


4.8570 


5-8284 


6.7998 


7.7712 


8. 7426 


1.3809 


3i 


0.9713 


1.9426 


2.9139 


3-8852 


4.8565 


5.8278 


6.7991 


7.7704 


8.7417 


1.380S 


32 


0.9712 


1.9424 


2.9136 


3.8848 


4.8560 


5.8272 


6.7984 


7 7697 


8.7409 


1.3S08 


33 


0.9711 


1.9422 


2.9133 


3.8844 


4.8556 


5.8267 


6.7978 


7.7689 


8. 7400 


1.3S07 


34 


0.9710 


1.9420 


2.9130 


3.8841 


4.855I 


5.8261 


6.7971 


7.7681 


8- 739i 


1.3S06 


35 


0.9709 


1. 9418 


2.9128 


3-8837 


4.8546 


5.S255 


6.7964 


7-7674 


8. 7383 


1.3806 


36 


0.9708 


I.94I7 


2.9125 


3.8833 


4.8541 


5-8250 


6.795S 


7.7666 


8. 7374 


1.3805 


37 


0.9707 


1. 9415 


2.9122 


3.8829 


4.8537 


5-8244 


6.7951 


7-7658 


8. 7366 


1.3804 


38 


0.9706 


I-94I3 


2.91 19 


3-8825 


4.8532 


5.8238 


6.7944 


7-7651 


8-7357 


1.3S04 


39 


0.9705 


1.9411 


2.9116 


3.8S22 


4-8527 


5-8232 


6.7938 


7-7643 


8-7349 


1.3S03 


40 


0.9704 


1.9409 


2.9113 


3.8S18 


4.8522 


5.8227 


6.7931 


7-7636 


8.7340 


1.3S02 


4i 


0.9703 


1.9407 


2.9110 


3.8814 


4.8517 


5.8221 


6.7924 


7.7628 


8-7331 


1.3S02 


42 


0.9703 


1.9405 


2.9108 


3.8810 


4-S5I3 


5-8215 


6.791S 


7.7620 


8.7323 


1. 380 1 


43 


0.9702 


1.9403 


2.9105 


3.8806 


4.8508 


5.8209 


6. 791 1 


7.7612 


S.73I4 


1.3800 


44 


0.9701 


1. 9401 


2.9102 


3.8802 


4-S503 


5-8203 


6.7904 


7.7604 


8.7305 


1-3799 


45 


0.9700 


1-9399 


2.9099 


3.8798 


4.8498 


5.8198 


6.7897 


7-7597 


8. 7296 


1-3799 


46 


0.9699 


1-9397 


2.9096 


3.8794 


4.8493 


5.8192 


6.7S90 


7-75S9 


8.72SS 


I-379S 


47 


0.9698 


1-9395 


2.9093 


3.879I 


4.8488 


5.81S6 


6.7SS4 


7.75SI 


8.7279 


1-3797 


48 


0.9697 


1-9393 


2.9090 


3.8787 


4.84S3 


5.81S0 


6.7877 


7-7573 


S.7270 


1-3797 


49 


0.9696 


1. 939i 


2.9087 


3.S7S3 


4.8479 


5.8i74 


6.7870 


7-7566 


S.7261 


1-3796 


50 


0.9695 


1.9389 


2.90S4 


3.8779 


4.8474 


5.S16S 


6.7863 


7-7558 


8.7253 


1-3795 


5i 


0.9694 


1.9388 


2.9081 


3-8775 


4.8469 


5.8163 


6.7856 


7-7550 


S.7244 


1-3795 


52 


0.9693 


1.93S6 


2.9078 


3-877I 


4.8464 


5-8157 


6.7S49 


7-7542 


S.7235 


1-3794 


53 


0.9692 


1-9384 


2.9075 


3.8767 


4.8459 


5.8i5i 


6.7S43 


7-7534 


S.7226 


1-3793 


54 


0.9691 


1.9382 


2.9072 


3.8763 


4.S454 


5.8I45 


6.7S36 


7-7526 


S.7217 


1-3792 


55 


0.9690 


1.9380 


2.9069 


3.8759 


4.S449 


5-Si39 


6.7S29 


7-7519 


S.720S 


1.3792 


56 


0.9689 


I.9378 


2.9066 


3.S755 


4.S444 


5.8133 


6.7S22 


7- 75i 1 


S.7199 


i-379i 


57 


0.9688 


1.9376 


2.9064 


3.8751 


4.S439 


5.8127 


6.7815 


7.7503 


S.7191 


1 -3790 


58 


0.9687 


1-9374 


2.9061 


3.8747 


4.8434 


5.S121 


6.780S 


7-7495 


S.71S2 


I-37S9 


59 


0.96S6 


1.9372 


2.905S 


3.8744 


4.8429 


5-Sn5 


6.7S01 


7.74S7 


S.7I73 


1.37S9 


60 


0.9685 


I-9370 


2.9055 


3.8740 


4.S424 


5.S109 


6-7794 


7-7479 


8.7164 


1.37S8 



9° HEIGHTS. 


107 


1 


2 


3 


4 


5 


G 


7 


8 


9 


b 


/ 
00 


CI543 


0.3086 


0.4629 


0.6172 


o.77i5 


o.9257 


1.0800 


1-2343 


1.3886 


0.2190 


0.1546 


0.3091 


0.4637 


0.6183 


0.7729 


0.9274 


1.0820 


1.2365 


1.3911 


0.2194 


01 


0.1548 


0.3097 


0.4645 


0.6194 


0. 7742 


0.9290 


1.0839 


J- 2387 


1.3936 


0.2198 


02 


0.1551 


0.3102 


0.4654 


0.6205 


0.7756 


0.9307 


1.0858 


1. 2410 


1.3961 


0.2202 


03 


0.1554 


0.3108 


0.4662 


0.6216 


0.7770 


0.9324 


1.0878 


1.2432 


1.3986 


0.2206 


04 


0.1557 


0.3113 


0.4670 


0.6227 


0.7784 


0.9340 


1.0897 


1.2454 


1. 4010 


0.2210 


05 


o.i559 


0.31 19 


0.4678 


0.6238 


0.7797 


o.9357 


1. 0916 


1.2476 


I.4035 


0.2214 


06 


0.1562 


0.3124 


0.4687 


0.6249 


0.7811 


0-9373 


I.C93 6 


1.2498 


1.4060 


0.2218 


07 


0.1565 


0.3130 


0.4695 


0.6260 


0.7825 


0.9390 


1.0955 


1.2520 


1.4085 


0.2222 


08 


0.1568 


0.3136 


0.4703 


0.6271 


0.7839 


0.9407 


1.0975 


1.2542 


1.4110 


0.2226 


09 


0.1571 


0.3141 


0.4712 


0.6282 


0.7853 


0.9423 


1.0994 


1.2564 


I.4I35 


0.2230 


10 


o.i573 


0.3I47 


0.4720 


0.6293 


0.7866 


0.9440 


1.1013 


1.2586 


1. 4160 


0.2234 


11 


0.1576 


0.3152 


0.4728 


0.6304 


0.7880 


0.9456 


1. 1032 


1.2608 


1.4184 


0.2238 


12 


o.i579 


0.3158 


0.4736 


0.6315 


0.7894 


0-9473 


1. 1052 


1.2630 


1.4209 


0.2242 


13 


0. 1582 


0.3163 


o.4745 


0.6326 


0. 7908 


0.9489 


1.1071 


1.2652 


L4234 


0.2246 


14 


0.1584 


0.3169 


o.4753 


0.6337 


0. 7922 


0.9506 


1. 1090 


1.2674 


1.4259 


0.2250 


15 


0.1587 


0.3I74 


0.4761 


0.6348 


C7935 


0.9523 


I.IIIO 


1.2697 


1.4284 


0.2254 


16 


0.1590 


0.3180 


0.4769 


0.6359 


o.7949 


o.9539 


1.1129 


1.2719 


1.4308 


0.2258 


!7 


o.i593 


0.3185 


0.4778 


0.6370 


0.7963 


o.9556 


1.1148 


1. 2741 


1-4333 


0.2262 


18 


o.i595 


0.3191 


0.4786 


0.6381 


0.7977 


o.9572 


1.1167 


1.2763 


1-4358 


0.2266 


19 


0.1598 


0.3196 


o.4794 


0.6392 


0.7991 


0.9589 


1.1187 


1.2785 


1.4383 


0.2270 


20 


0.1601 


0.3202 


0.4802 


0.6403 


0.8004 


0.9605 


1. 1206 


1.2807 


1.4407 


0.2274 


21 


0. 1604 


0.3207 


0.481 1 


0.6414 


0.8018 


0.9622 


1. 1225 


1.2829 


1.4432 


0.2278 


22 


0.1606 


0.3213 


0.4819 


0.6425 


0.8032 


0.9638 


1. 1244 


1. 2851 


1-4457 


0.2282 


23 


0.1609 


0.3218 


0.4827 


0.6436 


0.8046 


0.9655 


1. 1264 


1.2873 


1.4482 


0.2287 


24 


0.1612 


0.3224 


0.4835 


0.6447 


0.8059 


0.9671 


1. 1283 


1.2895 


1.4506 


0.2291 


25 


0.1615 


0.3229 


0.4844 


0.6458 


0.8073 


0.9688 


1. 1302 


1. 2917 


I-453I 


0.2295 


26 


0.1617 


0.3235 


0.4852 


0.6469 


0.8087 


0.9704 


1.1321 


1.2939 


1.4556 


0.2299 


27 


0.1620 


0.3240 


0.4860 


0.6480 


o.8ico 


0.9721 


1.1341 


1. 2961 


1.4581 


0.2303 


28 


0. 1623 


0.3246 


0.4868 


0.6491 


0.8114 


o.9737 


1. 1360 


1.2983 


1.4605 


0.2307 


29 


0. 1626 


0.3251 


0.4877 


0.6502 


0.8128 


o.9754 


1. 1379 


1.3005 


1.4630 


0.231 1 


30 


0. 1628 


o.3257 


0.4885 


0.6513 


0.8142 


0.9770 


1. 1398 


1.3027 


1.4655 


0.2315 


3 1 


0.1631 


0.3262 


0.4893 


0.6524 


0.8155 


0.9787 


1.1418 


1.3049 


1.4680 


0.2319 


32 


0.1634 


0.3268 


0.4901 


0.6535 


0.8169 


0.9803 


i- 1437 


1.3071 


1.4704 


0.2323 


33 


0.1637 


0.3273 


0.4910 


0.6546 


0.8183 


0.9819 


1. 1456 


1.3092 


1.4729 


0.2327 


34 


0.1639 


0.3279 


0.4918 


0.6557 


0.8196 


0.9836 


i- 1475 


1.3114 


1-4754 


0.2331 


35 


0. 1642 


0.3284 


0.4926 


0.6568 


0.8210 


0.9852 


1. 1494 


i.3!36 


1.4778 


0.2335 


36 


0.1645 


0.3290 


o.4934 


0.6579 


0.8224 


0.9869 


1.1514 


1.31S8 


1.4803 


0.2339 


37 


0. 1648 


0.3295 


0.4943 


0.6590 


0.8238 


0.9885 


1. 1533 


1. 3180 


1.4828 


0.2343 


38 


0. 1650 


0.3301 


0.4951 


0.6601 


0.8251 


0.9902 


1. 1552 


1.3202 


1.4853 


0.2347 


39 


0.1653 


0.3306 


0.4959 


0.6612 


0.8265 


0.9918 


1.1571 


1.3224 


1.4877 


0.2351 


40 


0. 1656 


0.33H 


0.4967 


0.6623 


0.8279 


0.9934 


1. 1590 


1.3246 


1.4901 


0.2355 


4i 


0.1658 


o.33i7 


C4975 


0.6634 


0.8292 


0.9951 


1. 1609 


1.3268 


1.4926 


0.2359 


42 


0.1661 


0.3322 


0.4984 


0.6645 


0.8306 


0.9967 


1. 1629 


1.3290 


I.495I 


0.2363 


43 


0. 1664 


0.3328 


0.4992 


0.6656 


0.8320 


0.9984 


1.1648 


1.3312 


1.4976 


0.2367 


44 


0. 1667 


o.3333 


0.5000 


0.6667 


0.8334 


1. 0000 


1. 1667 


1-3334 


1.5000 


0.2371 


45 


0.1669 


0-3339 


0.5008 


0.6678 


0.8347 


1. 0016 


1. 1686 


1-3355 


1.5025 


0.2375 


46 


0.1672 


0.3344 


0.5017 


0.6689 


0.8361 


1.0033 


1-1705 


1-3378 


1.5050 


0.2379 


47 


0.1675 


0.3350 


0.5025 


0.6700 


0.8375 


T.0049 


1. 1724 


1-3399 


1.5074 


0.2383 


48 


0.1678 


o.3355 


0.5033 


0.671 1 


0.8388 


1.0066 


i- 1743 


i-342i 


1.5098 


0.2387 


49 


0.1680 


0.3361 


0.5041 


0.6722 


0.8402 


1.0082 


1. 1763 


1-3443 


1.5123 


0.2391 


50 


0.1683 


0.3366 


0.5049 


0.6732 


0.8416 


1.0099 


1. 1782 


1-3465 


1.5148 


0.2395 


5i 


0.1686 


0.3372 


0.5057 


0.6743 


0.8429 


1.0115 


1.1801 


I-3487 


1-5172 


0.2399 


52 


0. 1689 


o.3377 


0.5066 


0.6754 


0.8443 


1. 0132 


1. 1820 


1.3509 


I.5I97 


0.2403 


53 


0.169 1 


o.3383 


0.5074 


0.6765 


0.8457 


1. 0148 


1. 1839 


I-353I 


1.5222 


0.2407 


54 


0.1694 


0.3388 


0.5082 


0.6776 


0.8470 


1. 0165 


1. 1859 


1-3553 


1.5247 


0.241 1 


55 


0. 1697 


o.3394 


0.5090 


0.6787 


0.8484 


1.0181 


1. 1878 


1-3574 


1.5271 


0.2415 


56 


0.1700 


0.3399 


0.5099 


0.6798 


0.8498 


1. 0197 


1. 1897 


I-3596 


1.5296 


0.2419 


57 


0.1702 


0.3404 


0.5107 


0.6809 


0.851 1 


1.0213 


1.1916 


1.3618 


1.5320 


0.2423 


58 


0.1705 


0.3410 


0.5115 


0.6820 


0.8525 


1.0230 


1. 1935 


1.3640 


1-5345 


0.2427 


59 


0.1708 


o.34i5 


0.5123 


0.6831 


0.8539 


1.0246 


1. 1954 


1.3662 


1-5369 


0.2431 


60 



108 


DISTANCES. 10° 


/ 
oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 

8.7164 


a 


0.9685 


I.9370 


2.9055 


3.8740 


4.8424 


5.8109 


6.7794 


7-7479 


1.3788 


OI 


0.9684 


1.9368 


2.9052 


3.8736 


4.8419 


5-8103 


6.7787 


7.7471 


8.7155 


L3787 


02 


0.9683 


1.9366 


2.9049 


3.8732 


4.8414 


5.8097 


6. 7780 


7.7463 


8.7146 


1.3786 


03 


0.9682 


1.9364 


2.9046 


3.8728 


4.8409 


5.8091 


6-7773 


7-7455 


8.7137 


1.3786 


04 


0.0681 


1.9362 


2.9043 


3-8724 


4.8404 


5.8085 


6.7766 


7-7447 


8.7128 


1.3785 


C5 


0.9680 


1.9360 


2.9040 


3.8720 


4.8399 


5.8079 


6.7759 


7-7439 


8.7119 


1-3784 


06 


0.9679 


1.9358 


2.9037 


3.8716 


4.8394 


5-8073 


6.7752 


7-743 1 


8.7110 


1.3783 


07 


0.9678 


I.9356 


2.9034 


3.8712 


4.8389 


5.8067 


6-7745 


7.7423 


8.7101 


1.3783 


08 


0.9677 


1-9354 


2.9031 


3-8707 


4.8384 


5.8061 


6.7738 


7.74I5 


8.7092 


1.3782 


09 


0.9676 


1-9352 


2.9028 


3-8703 


4.8379 


5-8055 


6.7731 


7.7407 


8.7083 


i.378i 


10 


0.9675 


I-9350 


2.9025 


3.8699 


4.8374 


5.8049 


6.7724 


7-7399 


8.7074 


1.3780 


11 


0.9674 


1.9348 


2.9022 


3-8695 


4.8369 


5-8043 


6.7717 


7-7391 


8.7065 


1.3780 


12 


0.9673 


1.9346 


2.9019 


3.8691 


4.8364 


5-8037 


6.7710 


7.7383 


8.7056 


1-3779 


13 


0.9672 


1-9344 


2.9015 


3.8687 


4-8359 


5.8031 


6.7703 


7-7375 


8.7046 


1.3778 


14 


0.9671 


1-9342 


2.9012 


3.8683 


4-8354 


5.8025 


6.7696 


7.7366 


8.7037 


1-3777 


15 


0.9670 


I.9340 


2.9009 


3.8679 


4.8349 


5.8019 


6.7689 


7-7358 


8.7028 


1.3776 


16 


0.9669 


1.9338 


2.9006 


3.8675 


4.8344 


5.8011 


6.7681 


7-7350 


8. 7019 


L3776 


17 


0.9668 


I.9336 


2.9003 


3.8671 


4.8339 


5.8007 


6.7674 


7-7342 


8.7010 


1-3775 


18 


0.9667 


1-9333 


2.5000 


3.8667 


4.8334 


5.8000 


6.7667 


7-7334 


8.7001 


1-3774 


I 9 


0.9666 


I-933I 


2.8997 


3.8663 


4.8329 


5-7994 


6.7660 


7.7326 


8.6991 


1-3773 


20 


0.9665 


1.9329 


2.8994 


3-8659 


4.8324 


5.7988 


6.7653 


7.73i8 


8.6982 


1-3773 


21 


0.9664 


I.9327 


2.8991 


3.8655 


4.8318 


5-7982 


6. 7646 


7.7309 


8.6973 


1-3772 


22 


0.9663 


L9325 


2.8988 


3.8651 


4.8313 


5-7976 


6.7638 


7-730I 


8.6964 


I.377I 


23 


0.9662 


L9323 


2.8985 


3.8646 


4.8308 


5-7970 


6.7631 


7.7293 


8.6954 


I-3770 


24 


0.9661 


1.9321 


2.8982 


3.8642 


4-8303 


5-7963 


6.7624 


7.7285 


S.6945 


1-3769 


25 


0.9660 


I.93I9 


2.8979 


3.8638 


4.8298 


5.7957 


6.7617 


7.7276 


8.6936 


I-37C9 


26 


0.9659 


r-93i7 


2.8976 


3.8634 


4.8293 


5-7951 


6.7610 


7.7268 


8.6927 


1.376S 


27 


0.9657 


I-93I5 


2.8972 


3.8630 


4.8287 


5-7945 


6.7602 


7.7260 


8.6917 


I-3767 


28 


0.9656 


I.93I3 


2.8969 


3.8626 


4.8282 


5-7939 


6-7595 


7-7252 


8.650S 


1.3766 


29 


0.9655 


1. 931 1 


2.8966 


3.8622 


4.8277 


5-7932 


6.7588 


7.7243 


8.6859 


L3765 


3o 


0.9654 


1.9309 


2.8963 


3.8617 


4.8272 


5.7926 


6.7581 


7.7235 


8.6S89 


L3765 


3i 


0.9653 


1.9307 


2.8960 


3.8613 


4.8267 


5-7920 


6-7573 


7.7227 


S.6880 


' 1.3764 


32 


0.9652 


I.9305 


2.8957 


3.8609 


4.8261 


5-7914 


6.7566 


7.7218 


8.6870 


, 1-3763 


33 


0.9651 


1.9302 


2.8954 


3.8605 


4.8256 


5-7907 


6-7559 


7.7210 


8.6861 


1.3762 


34 


0.9650 


r.9300 


2.8951 


3.8601 


4.8251 


5-790I 


6.7551 


7.7201 


S.6852 


1.3761 


35 


0.9649 


1.9298 


2.8947 


3-8597 


4.8246 


5-7S95 


6-7544 


7.7I93 


S.6S42 


i.376i 


36 


0.9648 


1.9296 


2.8944 


3.8592 


4.8240 


5-7888 


6.7537 


7.7185 


S.68^3 


1.3760 


37 


0.9647 


1.9294 


2.8941 


3.S588 


4-8235 


5.78S2 


6.7529 


7.7176 


8.6S23 


1-3759 


38 


0.9646 


1.9292 


2.8938 


3-S584 


4.S230 


5-7876 


6.7522 


7.7168 


8.6S14 


1-3759 


39 


0.9645 


1.9250 


2.8935 


3-8580 


4.8225 


5.7870 


6.7515 


7-7159 


8.6S04 


1.3758 


40 


0.9644 


1.9288 


2.8932 


3.8576 


4.8219 


5.7863 


6.7507 


7.7i5i 


8.6795 


1-3757 


4i 


0.9643 


1.9286 


2.8928 


3.8571 


4.8214 


5.7857 


6.7500 


7.7I43 


8.6785 


L3756 


42 


0.9642 


1.9284 


2.8925 


3.8567 


4.8209 


5.7851 


6.7492 


7.7I34 


S.6776 


1-3755 


43 


0.9641 


1.9281 


2.S922 


3.8563 


4.8203 


5-7844 


6.7485 


7.7126 


8.6766 [ 


1-3755 


44 


0.9640 


1.9279 


2.S919 


3.S558 


4.8198 


5.7838 


6.7477 


7-7II7 


8.6757 


1-3754 


45 


0.9639 


1.9277 


2.8916 


3-8554 


4.SI93 


5.7S31 


6.7470 


7.7108 


8.6747 


1-3753 


46 


0.9638 


i.9 2 75 


2.8912 


3-8550 


4.8187 


5-7S25 


6. 7462 


7.7100 


8.6737 ' 


1.3752 


47 


0.9636 


i.9 2 73 


2.8909 


3-S546 


4.8182 


5.7S19 


6.7455 


7.7091 


8.672S 


1-3752 


48 


0.9635 


1.9271 


2.8506 


3.S54I 


4.8177 


5-7Si2 


6.7448 7-7083 


S.6718 


I-375I 


49 


0.9634 


1.9269 


2.8903 


3-8537 


4.8172 


5.7S06 


6.7440 7-7074 


S.6709 


I-3750 


50 


0.9633 


1.9266 


2.S900 


3-8533 


4.8166 


5-7799 


6.7433 


7.7066 


S.6699 


1-3749 


5i 


0.9632 


1.9264 


2.8896 


3-8529 


4.S161 


5-7793 


6.7425 


7.7C57 


S.66S9 


i.374a 


52 


0.9631 


1.9262 


2.S893 


3-8524 


4-SI55 


5.77S6 


6.7417 


7.7049 


8.6680 


1.3748 


53 


0.9630 


1.9260 


2.8890 


3-8520 


4.8150 


5.77SO 


6.7410 


7.7040 


S.6670 


1-3747 


54 


0.9629 


1.925S 


2.8887 


3-35i6 


4.Si45 


5-7773 


6. 7402 


7-7031 


8.6660 


I.3746 


55 


0.9628 


1.9256 


2.8SS3 


•3-85" 


4.8139 


5-7767 


6-7395 


7-7023 


s.6650 : 


1 -3 745 


56 


0.9627 


1.9254 


2.8S80 


3-S5Q7 


4-Si34 


5.776i 


6.73S7 


7.7014 


s.6641 


1-3744 


57 


0.9626 


1.9251 


2.8877 


3-8503 1 


4.S12S 


5-7754 


6.7380 


7.7005 


8.6631 


1-3744 


53 


0.9625 


1.9249 


2.S874 


3.849S 1 


4.S123 


5.7748 


6.7372 


7.6997 


s.6621 


1-3743 


59 


0.9624 


1.9247 


2.8871 


3.8494 , 


4.S11S 


5-7741 


6.7365 7.69SS 


8.6612 


1.3742 


60 


0.9622 


1.9245 


2.8867 


3.8490 j 


4.S112 


5-7735 


6.7357 | 7.6979 


s.6602 


1-3742 



10° HEIGHTS. 






109 


1 


2 


3 


4 


5 


6 


7 


8 


9 


b 


00 


0.1708 


o.34i5 


0.5123 


0.6831 


o.8539 


1.0246 


1 -1954 


1.3662 


L5369 


0.2431 


0.1710 


0.3421 


0.5131 


0.6842 


0.8552 


1.0262 


1. 1973 


1.3683 


1-5394 


0.2435 


01 


0.1 713 


0.3426 


0.5140 


0.6853 


0.8566 


1.0279 


1. 1992 


I.3705 


I-54I9 


0.2439 


02 


0.1716 


0-343 2 


0.5148 


0.6864 


0.8580 


1.0295 


I.2CII 


L3727 


J -5443 


0.2443 


03 


0.1719 


o.3437 


0.5156 


0.6875 


0.8593 


1.0312 


I.203O 


1-3749 


1.5468 


0.2447 


04 


0.1721 


0.3443 


0.5164 


0.6886 


0.8607 


1.0328 


I.2050 


L377I 


1-5493 


0.2451 


05 


0.1724 


0.3448 


o.5i73 


0.6896 


0.8620 


I-0345 


I.2069 


1-3793 


I-55I7 


0.2455 


c6 


c.1727 


0.3454 


0.5180 


0.6907 


0.8634 


1. 0361 


I.2088 


1.3814 


I-554I 


0.2459 


07 


0.1730 


o.3459 


0.5189 


0.6918 


0.8648 


I-0377 


I. 2IO7 


1.3836 


1.5566 


0.2463 


08 


0.1732 


0.3464 


o.5i97 


0.6929 


0.8661 


1 -0393 


1. 2126 


1.3858 


I.5590 


0.2467 


09 


O.I735 


0.3470 


0.5205 


0.6940 


0.8675 


1. 0410 


I.2I45 


1.3880 


1. 5615 


0.2471 


10 


0.1738 


o.3475 


c.5213 


0.6951 


0.8688 


1.0426 


1. 2164 


1.3902 


L5639 


0.2475 


11 


0.1740 


0.3481 


0.5221 


0.6962 


0.8702 


1.0442 


1. 2183 


1.3923 


1.5664 


0.2479 


12 


O.I743 


0.3486 


0.5229 


0.6973 


0.8716 


1.0459 


I.2202 


1-3945 


1.5688 


0.2483 


13 


0. 1 746 


0.3492 


0.5238 


0.6984 


0.8729 


I-0475 


1. 2221 


1.3967 


I-57I3 


0.2487 


14 


0.1749 


0-3497 


0.5246 


0.6994 


0.8743 


1.0492 


I.224O 


1.3989 


1-5737 


0.2491 


15 


0.1 751 


0.3503 


0.5254 


0.7005 


0.8757 


1.0508 


1.2259 


1. 4010 


1.5762 


c.2495 


16 


0.1754 


0.3508 


0.5262 


0.7016 


c.8770 


1.0524 


I.2278 


1.4032 


1.5786 


0.2499 


17 


0.1757 


o.35i3 


0.5270 


0. 7027 


c.8784 


1.0540 


I.2297 


1.4054 


1.5810 


0.2503 


18 


0.1759 


o.35i9 


0.5278 


0.7038 


0.8797 


I.0557 


I. 2316 


1.4076 


1.5835 


0.2507 


19 


0.1762 


0.3524 


c.5287 


0.7049 


0.881 1 


I.0573 


1-2335 


1.4098 


1.5860 


0.251 1 


20 


0.1765 


0.3530 


0.5295 


0.7060 


0.8824 


1.0589 


L2354 


1.4119 


1.5884 


0.2515 


21 


0.1768 


o.3535 


0.5303 


0.7070 


0.8838 


1.0606 


1-2373 


1.4141 


1.5908 


0.2519 


22 


0.1770 


o.354i 


o.53i 1 


0.7081 


0.8852 


1.0622 


I.2392 


1. 4162 


1-5933 


0.2523 


23 


0.1773 


0.3546 


o.53i9 


0. 7092 


0.8865 


1.0638 


I.24II 


1.4184 


1-5957 


0.2527 


24 


0.1776 


o.3552 


o.5327 


0.7103 


0.8879 


1.0655 


I.243O 


1.4206 


1.5982 


0.2531 


25 


0.1778 


o.3557 


o.5335 


0.71 14 


0.8892 


1. 0671 


I.2449 


1.4228 


1.6006 


0.2535 


26 


0.1 781 


0.3562 


0.5344 


0.7125 


0.8906 


1.0687 


I.2468 


1.4250 


1. 6031 


0.2539 


27 


0.1784 


0.3568 


0.5352 


0.7136 


0.8920 


1.0703 


I.2487 


1. 4271 


1.6055 


0.2543 


28 


0.1787 


o.3573 


0.5360 


0.7146 


0.8933 


1.0720 


I.2506 


1.4293 


1.6079 


0.2547 


29 


0.1789 


o.3579 


0.5368 


0.7I57 


0.8947 


1.0736 


1.2525 


i.43i4 


1. 6104 


0.2551 


3° 


0.1792 


0.3584 


o.5376 


0.7168 


0.8960 


1.0752 


I.2544 


I-4336 


1.6128 


0.2555 


3 1 


0.1795 


0.3590 


0.5384 


0.7179 


0.8974 


1.0769 


I.2563 


1.4358 


i.6i53 


0.2559 


32 


0.1797 


o.3595 


o.5392 


0.7190 


0.8987 


1.0785 


I.2582 


1.4380 


1.6177 


0.2563 


33 


0.1800 


0.3600 


0.5401 


0.7201 


0.9001 


1. 0801 


1. 260I 


1.4402 


1.6202 


0.2567 


34 


0. 1803 


0.3606 


C54C9 


0.7212 


0.9014 


1.0817 


I.262O 


1.4423 


1.6226 


c.2571 


35 


0.1806 


0.361 1 


o.54i7 


0. 7222 


0.9028 


1.0834 


I.2639 


I-4445 


1.6250 


0.2575 


36 


0.1808 


0.3617 


o.5425 


0.7233 


0.9041 


1.0850 


I.2658 


1.4466 


1.6275 


0.2579 


37 


0.1811 


0.3622 


o.5433 


0.7244 


0.9055 


1.0866 


I.2677 


1.4488 


1.6299 


0.2583 


38 


0.1814 


0.3627 


0.5441 


0.7255 


c.9069 


1.0882 


I.2696 


1.4510 


1.6323 


0.2587 


39 


0.1816 


0.3633 


0.5449 


0. 7266 


0.9082 


1-0898 


I.27I5 


I-453I 


1.6348 


0.2591 


40 


0.1819 


0.3638 


o.5457 


0.7276 


0.9096 


1.0915 


1-2734 


1-4553 


1.6372 


o.2595 


4i 


0.1822 


0.3644 


0.5465 


0.7287 


0.9109 


1.0931 


1-2753 


1-4574 


1.6396 


0.2599 


42 


0.1825 


0.3649 


o.5474 


0. 7298 


0.9123 


1.0947 


1.2772 


1.4596 


1. 6421 


0.2603 


43 


0.1827 


0.3654 


0.5482 


0.7309 


0.9136 


1.0963 


1.2790 


1. 4618 


1.6445 


0.2607 


44 


0. 1830 


0.3660 


0.5490 


0.7320 


0.9150 


1.0979 


1.2809 


1.4639 


1.6469 


0.261 1 


45 


0.1833 


0.3665 


0.5498 


0.7330 


0.9163 


1.0996 


1.2828 


1. 4661 


1.6493 


0.2615 


46 


0.1835 


0.3671 


0.5506 


0.734I 


0.9177 


1.1012 


1.2847 


1.4682 


1.6518 


0.2619 


47 


0.1838 


0.3676 


o.55i4 


0. 7352 


0.9190 


1. 1028 


1.2866 


1.4704 


1.6542 


0.2623 


48 


0. 1841 


0.3681 


0.5522 


0.7363 


0.9204 


1. 1044 


1.2885 


1.4726 


1.6566 


0.2627 


49 


0.1843 


0.3687 


0.5530 


o.7374 


0.9217 


1.1061 


1.2904 


1.4748 


1. 6591 


0.2631 


50 


0. 1846 


0.3692 


o.5538 


0.7384 


0.9231 


1. 1077 


1.2923 


1.4769 


1.6615 


0.2635 


5i 


0.1849 


0.3698 


o.5546 


o.7395 


0.9244 


1. 1093 


1.2942 


1.4790 


1.6639 


0.2639 


52 


0.1852 


0.3703 


o.5555 


0. 7406 


0.9258 


1.1109 


1. 2961 


1. 4812 


1-6664 


0.2643 


53 


0.1854 


0.3708 


0.5563 


0.7417 


0.9271 


1.1125 


1.2979 


1-4834 


1.6688 


0.2647 


54 


0.1857 


o.37i4 


o.557i 


0. 7428 


0.9285 


1.1141 


1.2998 


•1.4855 


1. 6712 


0.2651 


55 


0.1860 


o.37i9 


o.5579 


o.7438 


0.9298 


1.1158 


1.3017 


1.4877 


1.6736 


0.2655 


56 


0.1862 


0.3725 


0.5587 


o.7449 


0.9312 


1.1174 


1.3036 


1.4898 


1. 6761 


0.2659 


57 


0.1865 


0.3730 


o.5595 


0.7460 


0.9325 


1.1190 


1.3055 


1.4920 


1.6785 


0.2663 


58 


0.1868 


o.3735 


0.5603 


0.7471 


o.9339 


1. 1206 


1-3074 


1.4942 


1.6809 


0.2667 


59 


0.1870 


o.374i 


0.561 1 


0. 7482 


o.9352 


1. 1222 


1-3093 


1.4963 


1.6834 


0.2671 


60 



110 DISTANCES. 


11° 


oo 


1 


2 


3 


4 


5 


6 


7 


8 


9 


a 


0.9622 


1.9245 


2.8867 


3.8490 


4.8112 


5-7735 


6-7357 


7.6979 


8.6602 


1-3742 


OI 


0.9621 


L9243 


2.8864 


3.8485 


4.8107 


5.7728 


6.7349 


7.6971 


8.6592 


1-3741 


02 


0.9620 


1.9240 


2.8861 


3.8481 


4.8101 


5-7721 


6-7342 


7.6962 


8.6582 


! 1-3740 


03 


0.9619 


1.9238 


2.8857 


3-8477 


! 4.8096 


5.77I5 


6-7334 


7-6953 


8.6572 


1-3739 


04 


0.9618 


1.9236 


2.8854 


3-8472 


4.8090 


5.77o8 


6.7326 


7.6944 8.6562 


1.3738 


05 


0.9617 


1.9234 


2.8851 


3.8468 


4.8085 


5-7702 


6.7319 


7.6936 ! 8.6553 


! 1.3738 


06 


0.9616 


1.9232 


2.8848 


3-8463 


| 4.8079 


5.7695 


6. 731 1 


7.6927 | 8.6543 


1-3737 


07 


0.9615 


1.9230 


2.8844 


3-8459 


! 4.8074 


5.7689 


6.7303 


7.6918 i 8.6533 


I.3736 


08 


0.9614 


1.9227 


2.8841 


3-8455 


i 4.8068 


5.7682 


6. 7296 


7.6909 ! 8.6523 


! J -3735 


09 


0.9613 


1.9225 


2.8838 


3-8450 


4.8063 


5.7675 


6.7288 


7.6901 8.6513 


1-3734 


IC 


0.961 1 


1.9223 


2.8834 


3.8446 


4.8057 


5-7669 


6.7280 


7.6892 


8.6503 


1-3734 


II 


0.9610 


1. 9221 


2.8831 


3.8441 


4.8052 


5.7662 


6.7273 


7.6883 


8.6493 


1-3733 


12 


0.9609 


1. 9218 


2.8828 


3-8437 


4.8046 


5.7655 


6.7265 j 7.6874 i 8.6483 


1-3732 


13 


0.9608 


1. 9216 


2.8824 


3-8433 


4.8041 


5-7649 


6.7257 


7.6865 8.6473 


I-373I 


14 


0.9607 


1.9214 


2.8821 


3.8428 


4.8035 


5-7642 


6.7249 


7.6856 8.6463 


1-373° 


15 


0.9606 


1. 9212 


2.8818 


3.8424 


4.8030 


5.7635 


6.7241 


7.6847 ! 8.6453 


I.3730 


16 


0.9605 


1. 9210 


2.8814 


3-84I9 


4.8024 


5-7629 


6.7234 


7.6838 i 8.6443 


1-3729 


17 


0.9604 


1.9207 


2.8811 


3-84I5 


4.8018 


5.7622 


6. 7226 


7.6830 j 8.6433 


1.3728 


18 


0.9603 


1.9205 


2.8808 


3.8410 


4.8013 


5-76i5 


6.7218 


7.6821 ! 8.6423 


I.3727 


19 


0.9601 


1.9203 


2.8804 


3.8406 


4.8007 


5.7609 


6.7210 


7.6812 


8.6413 


1.3726 


20 


0.9600 


1. 9201 


2.8801 


3.8401 


4.8002 


5.7602 


6. 7202 


7.6803 


8.6403 


1.3726 


21 


0.9599 


1. 9198 


2.8798 


3-8397 


4.7996 


5-7595 


6.7195 


7-6794 


8.6393 


I.3725 


22 


0.9598 


1. 9196 


2.8794 


3-8392 


4.7990 


5-7589 


6.7187 


7.6785 


8.6383 


1-3724 


23 


Q-9597 


1. 9194 


2.8791 


3.8388 


4-7985 


5-7582 


6.7179 


7.6776 j 8.6373 


1-3723 


24 


0.9596 


1. 9192 


2.8788 


3-8383 


4-7979 


5-7575 


6.7171 


7.6767 ' 8.6363 


1.3722 


25 


o.9595 


1. 9189 


2.8784 


3-8379 


4-7974 


5.7568 


6.7163 


7.6758 i 8.6352 


1.3722 


26 


0.9594 


1.9187 


2.8781 


3-8374 


4.7968 


5.7562 


6.7155 


7.6749 8.6342 


! 1.3721 


27 


0.9592 


1.9185 


2.8777 


3-8370 


4.7962 


5-7555 


6.7147 


7.6740 8.6332 


1.3720 


28 


0.9591 


1.9183 


2.8774 


3-8365 


4-7957 


5.7548 


6.7139 


7.6731 8.6322 


I.37I9 


29 


0.9590 


1. 9180 


2.8771 


3-8361 


4-7951 


5.7541 


6.7131 


7.6722 8.6312 


1.3718 


3° 


0.95S9 


1.9178 


2.8767 


3-8356 


4.7945 


5-7534 


6.7124 


7.6713 8.6302 


1.3718 


31 


0.9588 


1.9176 


2.8764 


3-8352 


4.7940 


5-7528 


6.7116 


7.6704 


8.6291 


I-37I7 


3 2 


0.9587 


1.9174 


2.8760 


3-8347 


4-7934 


5-7521 


6.7108 


7.6694 8.6281 


i.37i6 


33 


0.9586 


1.9171 


2.8757 


3-8343 


4.7928 


5-7514 


6.7100 


7.6685 ; S.6271 


: I-37I5 


34 


0.9585 


1. 9169 


2.8754 


3.8338 


4-7923 


5-7507 


6. 7092 


7.6676 8. 6261 


I.37I4 


35 


0.9583 


1.9167 


2.8750 


3-^333 


4-79*7 


5-75O0 


6.7084 


7.6667 . 8.6250 


I-37I4 


36 


0.9582 


1. 9164 


2.8747 


3-8329 1 


4.7911 


5-7493 


6.7076 


7.6658 ; 8.6240 


I.37I3 


37 


0.9581 


1. 9162 


2.8743 


3-8324 j 


4-7905 


5.7487 


6.706S : 7.6649 : S.6230 


I-37I2 


38 


0.9580 


1. 9160 


2.8740 


3.8320 


4.7900 


5-74So 


6.7060 


7.6640 S.6219 


i-37" 


39 


o.9579 


1. 9158 


2.8736 


3.S3I5 1 


4-7S94 


5-7473 


6.7052 


7.6630 S.6209 


I-37IO 


40 


o.9578 


i^SS 


2.8733 


3-83II 


4.7888 


5.7466 


6.7044 


7.6621 


8.6199 


1.3710 


4i 


o.9577 


i.9 J 53 


2.8730 


3-8306 


4.7883 


5-7459 


6.7036 


7.6612 


S.6189 


1.3709 


42 


o.9575 


1.9151 


2.8726 


3-8301 


4-7877 


5-7452 


6.7027 


7.6603 


S.6178 


1.3708 


43 


o.9574 


1,9148 


2.8723 


3-8297 


4.7871 


5-7445 


6. 7019 


7.6593 


S.616S 


1.3707 


44 


o.9573 


1. 9146 


2.8719 


3.8292 


4.7865 


5-743S 


6. 701 1 


7.6584 


S.6157 


1.3706 


45 


0.9572 


1.9144 


2.8716 


3.82S7 


4.7S59 


5.7431 


6.7003 


7-6575 


8.6147 


1.3706 


46 


Q-957 1 


1.9141 


2.8712 


3.8283 


4-7854 


5.7424 


6.6095 


7.6566 


8.6136 


I.3705 


47 


0.9570 


*-9*39 


2.8709 


3.8278 


4.7S4S 


5.74I7 


6.69S7 


7-6556 


8.6126 


1.3704 


4 s 


0.9568 


i.9i37 


2.8705 


3.8274 


4.7S42 


5-74IO 


6.6979 


7-6547 


S.6116 


1-3703 


49 


0.9567 


i-9i34 


2.8702 


3.8269 


4.7S36 


5.7403 


6.6971 


7.6558 S.6105 


1.3702 


50 


0.9566 


1. 9132 


2.S69S 


3.S264 


4-7830 


5.7396 


6.6962 


7.6529 S.6o95 


1.3702 


5i 


0.9565 


1.9130 


2.8695 


3.8260 


4.7824 


5.7389 


6.6954 


7.6519 S.60S4 


1.3701 


52 


0.9564 


1. 9127 


2.S691 


3-S255 ! 


4.7S19 


5-73S2 


6.6946 j 7.6510 S.6073 | 


1.3700 


53 


0.9563 


1.9125 


2.S68S 


3-8250 


4-7Si3 


5-7375 


6.693S 7.6500 S.6063 : 


1.3699 


54 


0.9561 


1.9123 


2.S684 


3.8245 


4.7807 


5.736S 


6-6930 


7.6491 S.6052 


1.3698 


55 


0.9560 


1. 9120 


2.S6S1 


3.8241 | 


4.7801 


5-736i 


6.6921 


7.64S2 S.6042 


1.369S 


56 


o.9559 


1.911S 


2.S677 


3.S236 1 


4-7795 


5- 7354 


6.6913 


7.6472 8.6031 


1.3697 


57 


o.955S 


1.9116 


2.S674 


3.8231 1 


4-7789 


5-7347 


6.6905 


7.6463 S.6021 


1.3696 


58 


o.9557 


1.9113 


2.8670 


3.8227 


4-77S3 


5-7340 


6.6897 7.6453 8.6010 


1.3695 


59 


i 0.9556 


1.9111 


2.S667 


3.8222 


4*7778 


5-7333 


6.6SS9 7.6444 S.600O 


1.3694 


60 


I 0.9554 1.9109 


2.S663 


3.S217 


4.7772 


5.7326 6.6SS0 7-6435 8-5989 


1.3694 



11° HEIGHTS. Ill 


1 


2 


3 

0.561 1 


4 


5 


6 


7 


8 


9 


b 


00 


0.1870 


o.374i 


0.7482 


o.9352 


1. 1222 


1-3093 


1.4963 


1.6834 


0.2671 


0.1873 


0.3746 


0.5619 


0.7492 


0.9366 


1. 1239 


1.3112 


1.4985 


1.6858 


0.2675 


01 


0.1876 


0.3752 


0.5627 


0.7503 


o.9379 


I- 1255 


1-3131 


1. 5006 


1.6882 


0.2679 


02 


0.1878 


o.3757 


0.5635 


0.75M 


0.9392 


1.1271 


1. 3i49 


1.5028 


1.6906 


0.2683 


03 


0.1881 


0.3762 


0.5644 


0.7525 


0.9406 


1. 1287 


1.3168 


1.5050 


1. 6931 


0.2687 


04 


0. 1884 


0.3768 


0.5652 


0.7536 


0.9420 


1. 1303 


1.3187 


1-5071 


1.6955 


0.2691 


05 


0. 1887 


o.3773 


0.5660 


0.7546 


o-9433 


1.1319 


1.3206 


I.5093 


1.6979 


0.2695 


06 


0.1889 


0.3778 


0.5668 


0.7557 


0.9446 


1. 1335 


1.3224 


1.5114 


1.7003 


0.2699 


07 


0. 1892 


0.3784 


0.5676 


0.7568 


0.9460 


I.I35I 


1.3243 


I.5I35 


1.7027 


0.2703 


08 


0.1895 


0.3789 


0.5684 


o.7578 


o.9473 


1. 1368 


1.3262 


I.5I57 


1. 7051 


0.2707 


09 


0.1897 


o.3795 


0.5692 


0.7589 


0.9487 


1. 1384 


1.3281 


1.5178 


1.7076 


0.2711 


10 


0.1900 


0.3800 


0.5700 


c.7600 


0.9500 


1. 1400 


1.3300 


1.5200 


1. 7100 


0.2715 


11 


0.1903 


0.3805 


0.5708 


0. 761 1 


0.9513 


1.1416 


1. 33i9 


1.5222 


1. 7124 


0.2719 


12 


0.1905 


0.381 1 


0.5716 


0. 7622 


0.9527 


1. 1432 


1.3338 


L5243 


I.7I49 


0.2723 


13 


0.1908 


0.3816 


o.57 2 4 


0.7632 


0.9540 


1. 1448 


1-3357 


1.5265 


i-7i7a 


0.2727 


14 


0.1911 


0.3821 


o.5732 


0.7643 


o.9554 


1. 1464 


1-3375 


1.5286 


1.7197 


0.2731 


15 


0.1913 


0.3827 


0.5740 


0.7654 


0.9567 


1. 1480 


1-3394 


I-5307 


1. 7221 


0.2735 


16 


0.1916 


0.3832 


o.5748 


0. 7664 


0.9580 


1. 1497 


I-34I3 


1.5329 


*.7245 


0.2739 


17 


0.1919 


0.3838 


0.5756 


0.7675 


o.9594 


I.I5I3 


1-3432 


I-5350 


1.7269 


0.2743 


18 


0.1921 


0.3843 


0.5764 


0.7686 


0.9607 


1. 1529 


i-345o 


1-5372 


L7293 


0.2747 


19 


0. 1924 


0.3848 


0.5773 


0.7697 


0.9621 


1. 1545 


1.3469 


1-5394 


I-73I7 


0.2751 


20 


0.1927 


0.3854 


0.5781 


0.7707 


0.9634 


1.1561 


1.3488 


I.54I5 


1. 7341 


0.2755 


21 


0.1930 


0.3859 


0.5789 


0.7718 


0.9648 


I-I577 


1-3507 


1.5436 


1.7366 


0.2759 


22 


0.1932 


0.3864 


o.5797 


0.7729 


0.9661 


1. 1593 


1-3525 


1-5458 


1.7390 


0.2763 


23 


o.i935 


0.3870 


0.5805 


0.7740 


0.9674 


1. 1609 


L3544 


1-5479 


I-74I4 


0.2767 


24 


0.1938 


0.3875 


0.5813 


0.7750 


0.9688 


1. 1625 


I-3563 


1.5500 


1.7438 


0.2771 


25 


0. 1940 


0.3880 


0.5821 


0.7761 


0.9701 


1.1641 


i.358i 


1-5522 


1.7462 


0.2775 


26 


0.1943 


0.3886 


0.5829 


0.7772 


0.9714 


1.1657 


1.3600 


1-5543 


1.7486 


0.2779 


27 


0. 1946 


0.3891 


0.5837 


0.7782 


0.9728 


1. 1674 


1-3619 


1-5565 


I-75IO 


0.2783 


28 


0. 1948 


0.3896 


0.5845 


o.7793 


0.9741 


1. 1689 


1-3637 


1.5586 


L7534 


0.2787 


29 


0.1951 


0.3902 


0.5853 


0.7804 


o.9755 


1-1705 


1-3656 


1.5607 


1.7558 


0.2791 


30 


0.1954 


0.3907 


0.5861 


0.7814 


0.9768 


1. 1722 


1-3675 


1.5629 


1.7582 


0.2795 


3 1 


0.1956 


0.39I3 


0.5869 


0.7825 


0.9781 


1.1738 


1.3694 


1.5650 


1.7607 


0.2799 


32 


0.1959 


0.3918 


0.5877 


0.7836 


o.9795 


1. * 753 


1.3712 


1-5671 


1.7630 


0.2803 


33 


0. 1962 


0.3923 


0.5885 


c.7846 


0.9808 


1. 1770 


I-373 1 


*.5693 


1.7654 


0.2807 


34 


0.1964 


0.3929 


0.5893 


0.7857 


0.9821 


1. 1786 


I.3750 


I.57I4 


1.7679 


0.281 1 


35 


0. 1967 


o.3934 


0.5901 


0.7868 


0.9835 


1. 1802 


1.3769 


I.5736 


1.7703 


0.2815 


36 


0.1970 


0.3939 


0.5909 


0.7878 


0.9848 


1.1818 


1.3787 


1-5757 


1.7727 


0.2819 


37 


0.1972 


o.3945 


o.59i7 


0.7889 


0.9861 


1. 1834 


1.3806 


1.5778 


1. 7751 


0.2823 


38 


0.1975 


0.3950 


o.5925 


0.7900 


0.9875 . 


1. 1850 


1.3825 


1.5800 


1-7775 


0.2827 


39 


0.1978 


o.3955 


o.5933 


0.7910 


0.9888 


1. 1866 


1.3843 


1.5821 


1.7798 


0.2831 


40 


0. 1980 


0.3961 


0.5941 


0.7921 


0.9901 


1. 1882 


1.3862 


1.5842 


1.7823 


0.2835 


4i 


0.1983 


0.3966 


o.5949 


0.7932 


0.9915 


1. 1898 


1.3881 


1.5864 


1.7847 


0.2839 


42 


0. 1986 


0.3971 


o.5957 


0.7942 


0.9928 


i.i9*4 


1.3899 


1.5885 


1. 7871 


0.2843 


43 


0.1988 


o.3977 


0.5905 


o.7953 


0.9941 


1. 1930 


1.3918 


1.5906 


1.7895 


0.2847 


44 


0.1991 


0.3982 


o.5973 


0.7964 


o.9955 


1. 1946 


1-3937 


1.5928 


1. 7919 


0.2851 


45 


0.1994 


0.3987 


0.5981 


o.7974 


0.9968 


1. 1962 


1-3955 


1-5949 


1.7942 


0.2855 


46 


0. 1996 


0.3993 


0.5989 


0.7985 


0.9981 


1. 1978 


1-3974 


1 -5970 


1.7966 


0.2859 


47 


0.1999 


0.3998 


0.5997 


0.7996 


0.9994 


I.I993 


1.3992 


i- 599i 


1.7990 


0.2863 


48 


0.2002 


0.4003 


0.6005 


o.8co6 


1.0008 


1. 2010 


1.4011 


1.6013 


1. 8014 


0.2867 


49 


0.2004 


0.4009 


0.6013 


0.8017 


1. 002 1 


1.2026 


1.4030 


1.6034 


1.8038 


0.2871 


50 


0.2007 


0.4014 


0.6021 


0.8028 


1.0034 


1. 2041 


1.4048 


1.6055 


1.8062 


0.2875 


5i 


0.2010 


0.4019 


0.6029 


0.8038 


1.0048 


1.2057 


1.4067 


1.6077 


i.£oS6 


0.2879 


52 


0.2012 


0.4024 


0.6037 


0.8049 


1. 0061 


1.2073 


1.4085 


1.6098 


1.8110 


0.2883 


53 


0.2015 


0.4030 


0.6045 


0.8060 


1.0075 


1.2089 


1. 4104 


1.6119 


1.8134 


0.2887 


54 


0.2018 


0.4035 


0.6053 


0.8070 


1.0088 


1. 2105 


1-4123 


1.6141 


1.8158 


0.2891 


55 


0.2020 


0.4040 


0.6061 


0.8081 


I.OIOI 


1.2121 


1.4141 


1. 6162 


1.8182 


0.2895 


56 


0.2023 


0.4046 


0.6069 


0.8092 


1.0114 


1.2137 


1.4160 


1.6183 


1.8206 


0.2899 


57 


0.2026 


0.4051 


0.6077 


0.8102 


1. 0128 


1. 2153 


1. 41 79 


1.6204 


1.8230 


0.2903 


58 


0.2028 


0.4056 


0.6085 


0.8113 


1.0141 


1. 2169 


1.4197 


1.6226 


1.8254 


0.2907 


59 


0.2031 


0.4062 


0.6092 


0.8123 


1. 01 54 


1.2185 


1. 4216 


1.6247 


1.8278 


0.291 1 


60 



112 DISTANCES. 12° 


/ 
oo 


1 


2 


3 


* 5 


6 


7 8 


9 

.8.5989 


a 


o.9554 


1. 9109 


2.8663 


3.8217 1 4.7772 
3.8213 :' 4.7766 


5-7326 


6.6880 7.6435 


1.3694 


OI 


o.9553 


1.9106 


2.8659 


5-7319 


6.6872 1 7.6425 


8.5978 


1-3693 


C2 


0.9552 


1.9104 


2.8656 


3.8208 4.7760 


5-7312 


6.6864 7.6416 


8.5968 


I-3693 


°3 


o.955i 


1. 9102 


2.8652 


3.8203 ; 4-7754 


5-7305 


; 6.6855 


7.6406 


8.5957 


1.3692 


04 


0.9550 


1.9099 


2.8649 


3.8198 , 4.7748 


5-7297 


6.6847 


7-6397 


8.5946 


1.3691 


05 


0.954S 


1.9097 


2.8645 


3.8194 ;; 4.7742 


5.7290 


6.6839 


7-6387 


8.5936 


1.3690 


06 


o.9547 


1.9094 


2.8642 


3.81S9 ;; 4.7736 


5-7283 


6.6830 


7.6378 


8.5925 


1.3689 


O? 


0.9546 


1.9092 


2.8638 


3.8184 II 4-773° 


5-7276 


6.6822 


7.6368 


8.5914 


1.36S8 


oS 


o.9545 


1.9090 


2.8634 


3.8179 || 4-7724 


5.7269 


6.6814 


7-6359 


8- 5903 


1.3687 


09 


0.9544 


1.9087 


2.8631 


3.8i75 


4.77i8 


5.7262 


6.6805 


7.6349 


8.5893 


1.3687 


10 


0.9542 


1.9085 


2.8627 


3.8170 


4.7712 


5.7255 


6.6797 


7.6340 


8.5882 


1.3686 


11 


0.9541 


1.9082 


2.S624 


3-8165 


4.7706 


5.7247 


6.6789 j 7.6330 


8.5871 


1.3685 


12 


0.9540 


1.9030 


2.8620 


3.8160 | 4.7700 


5.7240 


6.6780 1 7.6320 


8.5860 t 1.3684 


*3 


o.9539 


1.9378 


2.8616 


3.8155 :; 4.7694 


5-7233 


6.6772 ; 7-6311 


8.5S49 i 1.3683 


14 


0.9538 


1^075 


2.8613 


3.8150 4.768s 


5.7226 


6.6763 


, 7-6301 


8.5839 1.36S2 


15 


0.9536 


1-9073 


2.8609 


3.8146 ; 4.7682 


5-7219 


6.6755 


7.6291 


8.5828 | 1.36S1 


16 


0-9535 


1.9370 


2.86o5 


3.8141 4.7676 


5.72II 


6.6747 


7.6282 


8.5817 


1.36S1 


*7 


0.9534 


1.9068 


2.8602 


3.8136 ; 4.7670 


5-7204 


6.6738 


7.6272 


8.5806 


1.36E0 


18 


o.9533 


1.9066 


2.8598 


3.8131 , 4.7664 


5-7I97 


6.6730 


7.6262 


8-5795 


1.3679 


19 


0.9532 


1.9063 


2.8595 


3.8126 4.7658 


5-7I90 


6.6721 


7-6253 


8.5784 


1.3678 


20 


0.9530 


1. 9061 


2.S591 


3.8122 : 4.7652 

: 

3.81 1 7 4-7646 


5.7182 


6.6713 


7.6243 


8-5774 


1-3677 


21 


0.9529 


1.9058 


2.8588 


5-7175 


6.6704 


7.6233 


8.5763 


1.3676 


22 


0.9528 


1.9056 


2.8584 


3. 8l 12 j (4.7640 


5-7168 


6.6696 


7.6224 


8.5752 


1.3675 


2 3 


0.9527 


I.9353 


2.8580 


3.8107 j; 4.7634 


5.716O 


6.6687 


7.6214 


8.5741 


1-3674 


24 


0.9526 


1. 9051 


2.8577 


3.8102 4.7628 


5.7153 


6.6679 


7.6204 


8.5730 


I-3673 


2 5 


0.9524 


1.9049 


2.8573 


3.8097 I 4.7622 


5.7146 


6.6670 


7.6194 


8.5719 


1.3672 


26 


0.9523 


1.9046 


2.8569 


3.8092 j; 4.7615 


5-7I3S 


6.6662 


7.6185 


8.5708 


1.3672 


27! 


0.9522 


1.9044 


2.8566 


3-So87 I14-7609 


5-7I3I 


6.6653 


7-6175 


8.5697 


1.3671 


28! 


0.9521 


1. 9041 


2.8562 


3.S0S3 4.7603 


5.7124 


6.6644 


7.6165 


S.56S6 


1.3670 


29 


0.9519 


1.9039 


2.8558 


3.8078 | 4.7597 


5.7II7 


6.6636 


7-6155 


8.5675 


1.3669 


30 


0.9518 


1.9036 


2.8555 


3-8073 


4-759 1 


5.7IC9 


6.6627 


7.6146 


S.5664 


1.3668 


3i 


0.9517 


1.9034 


2.8551 


3.806S 


4.7585 


5.7I02 


6.6619 


7.6136 


8.5653 


1.3667 


32 


0.9516 


1.9031 


2.8547 


3.S0S3 


4-7579 


5-7094 


6.6610 


7.6126 


8.5642 


1.3667 


33 


0.9514 


1.9029 


2.8543 


3.805S 4.7572 


5-7087 


6.6601 


7.6116 


8.5630 


1.3666 


34; 


0.9513 


1.9027 


2.8540 


3.8053 j 4.7566 


5.70SO 


6.6593 


7.6106 


s.5619 ; 


1.3665 


35 


0.9512 


1.9024 


2.S536 


3.S04S 4.7560 


5-7072 


6.65S4 


7.6096 


s. 5 6o8j 


1.3664 


36 


0.9511 


1.9022 


2.8532 


3.8043 4-7554 


5-7055 


6.6575 


7.60S6 


8.5597 


1.3663 


37 


0.9510 


1. 9019 


2.S529 


3.S03S 4-7548 


5.7057 


6.6567 


7.6076 


8.5586 ! 


1.3662 


38 


0.9508 


1.9017 


2.S525 


3.8033 4-7542 


5.7050 


6.6558 


7.6066 


8.5575 


1.3661 


39 


0.9507 


1. 9014 


2.8521 


3.802S 


4-7535 


5.7042 


6.6550 


7-6057 


8.5564 : 


1.3660 


40 


0.9506 


1. 9012 


2.S51S 


3-8023 


4-7529 


5.7035 


6.6541 


7.6047 


8-5553 


1.3660 


4i 


0.9505 


1.9009 


2.8514 


3.8018 


4.7523 


5.7028 


6.6532 


7-6037 


8.5541 i 


1.3659 


42 


0.9503 


1.9007 


2.8510 


3.8013 


4-7517 


5.7O20 


6.6523 


7.6027 


S.5530!; 1 -365s 


43 


0.9502 


1.9004 


2.8506 


3.8008 [j 4.7510 


5.70I3 


6.6515 


7.6017 


S.5519 j 1.3657 


44 


0.9501 


1.9002 


2.8503 


3.8003 j 4.7504 


5.7O05 


6.6506 


7.6007 


S.550S 1.3656 


45 


0.9500 


1.S999 


2.S499 


3-7998 4-749S 


5.699S 


6.6497 


7-5997 


8.5406 1.3655 


46 


0.949S 


I.S997 


2.8495 


3-7993 ; 4-7492 


5.699O 


6.648S 


7o9S7 


S.54S5 


1.3654 


47 


0.9497 


1.S994 


2.8491 


3.7988 j 4.7485 


5.6983 


6.64S0 


7-5977 


8.5474 


I-3653 


4 s 


0.9496 


1.S992 


2.S4SS 


3-7983 


4-7479 


5.6975 


6.6471 


7-5967 


8.5463 


1.3652 


49 


0.9495 


1.8989 


2.S4S4 


3-7978 


4-7473 


5-696S 


6.6462 


7-5957 


8.545* 


1-3651 


50 


o.9493 


1.8987 


2.8480 


3-7973 


4.7467 


5.6960 


6.6453 


7-5947 


8.5440 


1-3651 


5i 


0.9492 


1.S9S4 


2.S476 


3.7968 


4.7460 


5-6952 


6.6444 


7-5937 


8.5429 


1.3650 


52 


0.9491 


1.8982 


2.S472 


3.7963 ; 4.7454 


5-6945 


6.6436 


7.5926 


S.5417 , 1,3649 


53^ 


0.9490 


1.S979 


2.8469 


3.795S ■ 4.744S 


5-6937 


6.6427 


7.59i6 


S.5406 j I.364S 


54' 


0.94S8 


1.8977 


2.8465 


3-7953 |i 4-7441 


5-6930 


6.6418 


7.5906 


8-5395 ! 1-3647 


55 


0.9487 


1.8974 


2.8461 


3.7948 |! 4.7435 


5.6922 


6.6409 


7.5806 


S.53S3 1-3646 


56 


0.94S6 


1.8971 


2.8457 


3-7943 i; 4.7429 


5.6914 


6.6400 


7. 5 SS6j 


8.5372 j 1.3645 


57 


0.9484 


1.8969 


2.8453 


3.793S 4.7422 


5.6907 


6.6391 


70S76 


8.5360 1.3644 


58: 


0.94S3 


1.S966 


2.8450 


3-7933 4-74i6 


5.6S99 


6.63S2 


7.5S66! 


S.5349 1.3643 


59 


0.9482 


1.S964 


2.8446 


3.792S 4.7410 


5.6892 


6.6374 


7.5S56 : 


8.533S 1.3642 


6o| 


0.94S1 


1. 8961 


2.8442 


3-79 2 3 4-7403 


5.6SS4 


6.6365 


7-5845 


8.5326 1. 3641 



12° 


HEIGHTS. 113 


1 


2 


3 


4 


5 


6 


7 


8 


9 


b 


00 


0.2031 


0.4062 


0.6092 


0.8123 


1.0154 


1. 2185 


1. 4216 


1.6247 


1.8278 


0.291 1 


0.2033 


0.4067 


0.6100 


0.8134 


1. 0167 


1. 2201 


1.4234 


1.6268 


1.8302 


0.2915 


01 


0.2036 


0.4072 


0.6108 


0.8144 


1.0181 


1. 2217 


1-4253 


1.6289 


1-8325 


0.2919 


02 


0.2039 


0.4078 


0.6116 


0.8155 


1. 0194 


1.2233 


1.4272 


1.6310 


1.8349 


0.2923 


03 


0.2041 


0.40S3 


0.6124 


0.8166 


1.0207 


1.2248 


1.4290 


1-6331 


1-8373 


0.2927 


04 


0.2044 


0.4088 


0.6132 


0.8176 


1. 0221 


1.2264 


1.4309 


I.6353 


1-8397 


0.2931 


05 


0.2047 


0.4093 


0.6140 


0.8187 


1.0234 


1.2280 


1-4327 


I.6374 


1.8420 


0.2935 


06 


0.2049 


0.4099 


0.6148 


0.8198 


1.0247 


1.2296 


1.4346 


I.6395 


1.8444 


0.2939 


07 


C.2052 


0.4104 


0.6156 


0.8208 


1.0260 


1. 2312 


1.4364 


1. 6416 


1.8468 


0.2943 


08 


0.2055 


0.4109 


0.6164 


0.8219 


1.0273 


1.2328 


1.4383 


1.6438 


1.8492 


0.2947 


09 


0.2057 


0.4115 


0.6172 


O.8229 


1.0287 


1.2344 


1. 4401 


1.6459 


1.8516 


0.2951 


10 


o.2o6o 


0.4120 


0.6180 


0.8240 


1.0300 


1.2360 


1.4420 


1.6480 


x.8540 


0.2955 


11 


0.2063 


0.4125 


0.6188 


0.8250 


x.0313 


1.2376 


1.4438 


1. 6501 


1.8564 


0.2959 


12 


0.2065 


0.4131 


0.6196 


0.8261 


1.0326 


1.2392 


1-4457 


1.6522 


1.8588 


0.2962 


13 


0.2068 


0.4136 


0.6204 


0.8272 


1.0340 


1.2408 


1-4475 


1-6543 


1. 8611 


0.2966 


14 


0.2071 


0.4141 


0.6212 


0.8282 


I.0353 


1.2424 


1.4494 


1-6565 


1.8635 


0.2970 


15 


0.2073 


0.4146 


0.6220 


0.8293 


1.0366 


1.2439 


1.4512 


1.6586 


1.8659 


0.2974 


16 


0.2076 


0.4152 


0.6227 


0.8303 


T.0379 


1.2455 


I-453I 


1.6607 


1.8682 


0.2978 


17 


0.2078 


0-4I57 


0.6235 


0.8314 


1.0392 


1. 2471 


1-4549 


1.6628 


1.8706 


0.2982 


18 


0.2081 


0.4162 


0.6243 


0.8324 


1.0406 


1.2487 


1.4568 


1.6649 


1.8730 


0. 2986 


i9 


0.2084 


0.4168 


0.6251 


0.8335 


1. 0419 


1.2503 


I-4587 


1.667c 


1-8754 


0.2990 


20 


c.2086 


0.4I73 


0.6259 


0.8346 


1.0432 


1. 2518 


1.4605 


1. 6691 


1.8778 


0.2994 


21 


0.2089 


0.4178 


0.6267 


0.8356 


1.0445 


1.2534 


1.4623 


1. 6712 


1.8801 


0.2998 


22 


0.2092 


0.4183 


0.6275 


0.8367 


1.0458 


1.2550 


1.4642 


1.6734 


1.8825 


0.5002 


23 


0.2094 


0.4189 


0.6283 


0.8377 


1.0472 


1.2566 


1.4660 


I-6755 


1.8849 


0.3006 


24 


0.2097 


0.4194 


0.6291 


0.8388 


1.0485 


1.2582 


1.4679 


1.6776 


1.8873 


0.3010 


25 


0.2100 


0.4199 


0.6299 


0.8398 


1.0498 


1.2598 


1.4697 


1.6797 


1.8896 


0.3014 


26 


0.2102 


0.4204 


0.6307 


0.8409 


1.0511 


1.2613 


I -47 I 5 


1. 6818 


1.8920 


0.3018 


27 


0.2105 


0.4210 


0.6315 


0.8420 


1.0524 


1.2629 


1-4734 


1.6839 


1.8944 


0.3022 


28 


0.2107 


0.4215 


0.6322 


0.8430 


I.0537 


1.2645 


1-4752 


1.6860 


1.8967 


0.3026 


29 


0.2110 


0.4220 


0.6330 


0.8440 


L055 1 


1. 2661 


1. 4771 


1. 6881 


1. 8991 


0.3030 


3° 


0.2113 


0.4226 


0.6338 


0.8451 


1.0564 


1.2677 


1.4790 


1.6902 


1.9015 


0.3034 


3 1 


0.2115 


0.4231 


0.6346 


0.8462 


I.0577 


1.2692 


1.4808 


1.6923 


1.9039 


0.3038 


32 


0.2118 


0.4236 


0.6354 


0.8472 


1.0590 


1.2^08 


1.4826 


1.6944 


1.9062 


0.3042 


33 


0.2121 


0.4241 


0.6362 


0.8483 


i.o£o3 


1.2724 


1.4845 


1.6965 


1.9086 


0.5046 


34 


0.2123 


0.4247 


0.6370 


0.8493 


1. 0616 


1.2740 


1.4863 


1.6986 


1.9110 


0.3050 


35 


0.2126 


0.4252 


0.6378 


0.8504 


1.0630 


1-2755 


1.4881 


1.7007 


i-9 J 33 


c.3054 


36 


0.2129 


0.4257 


0.6386 


0.8514 


1.0643 


1.2771 


1.4900 


1.7028 


i-9!57 


0.3058 


37 


0.2131 


0.4262 


0.6394 


0.8525 


1.0656 


1.2787 


1.4918 


1.7049 


1.9181 


0.3062 


38 


0.2134 


0.4268 


0.6401 


O.8535 


1.0669 


1.2803 


1-4937 


1.7070 


1.9204 


0.3066 


39 


0.2136 


0.4273 


0.6409 


0.8546 


1.0682 


T.2818 


1-4955 


1. 7091 


1.9228 


0.3070 


40 


0.2139 


0.4278 


0.6417 


O.8556 


1.0695 


1.2834 


1-4973 


1.7112 


1.9251 


0.3074 


41 


0.2142 


0.4283 


0.6425 


0.8567 


1.0708 


1.2850 


1.4992 


i-7 J 33 


I-9275 


0.3078 


42 


0.2144 


0.4289 


0.6433 


0.8577 


1. 0721 


1.2866 


1. 5010 


i-7 I 54 


1.9299 


0.3082 


43 


0.2147 


0.4294 


0.6441 


O.8588 


1 -0735 


1.2881 


1.5028 


I.7I75 


1.9322 


0.3086 


44 


0.2150 


0.4299 


0.6449 


0.8598 


1.0748 


1.2897 


1.5047 


1. 7196 


1.9346 


0.3090 


45 


0.2152 


0.4304 


0.6457 


0.8609 


1. 0761 


1.2913 


1.5065 


1. 7217 


1-9370 


0.3094 


46 


0.2155 


0.4310 


0.6464 


0.8619 


1.0774 


1.2929 


1.5084 


1.7238 


1-9393 


0.3098 


47 


0.2157 


0.43I5 


0.6472 


0.8630 


1.0787 


1.2944 


1-5102 


1.7259 


1. 941 7 


0.3102 


48 


0.2160 


0.4320 


0.6480 


O.8640 


1.0800 


1.2960 


1. 5120 


1.7280 


1.9440 


0.3106 


49 


0.2163 


0.4325 


0.6488 


0.8651 


1.0813 


1.2976 


I-5I38 


1. 7301 


1.9464 


0.3110 


50 


0.2165 


o.433i 


0.6496 


O.8661 


1.0826 


1.2992 


1-5*57 


1.7322 


1.9487 


0.3114 


5i 


0.2168 


o.433 6 


0.6504 


0.8671 


1.0839 


1.3007 


I.5I75 


1-7343 


1.9511 


0.3118 


52 


0.2170 


0.4341 


0.651 1 


0.8682 


1.0852 


1.3023 


I.5I93 


1.7364 


1-9534 


0.3121 


53 


0.2173 


0.4346 


0.6519 


0.8692 


1.0866 


I-3039 


1. 5212 


1.7385 


I.9558 


0.3125 


54 


0.2176 


Q-435 1 


0.6527 


0.8703 


1.0879 


I.3054 


1-5230 


1.7406 


1.9581 


0.3129 


55 


0.2178 


o.4357 


0.6535 


0.8713 


1.0892 


1.3070 


1.5248 


1.7427 


1.9605 


o.3i33 


56 


c.2181 


0.4362 


0.6543 


0.8724 


1.0905 


1.3186 


1.5267 


1.7448 


1.9629 


0.3137 


57 


0.2184 


0.4367 


0.6551 


0.8734 


1. 0918 


1.3101 


1.5285 


1.7468 


1.9652 


0.3141 


58 


0.2186 


0.4372 


0.6559 


0.8745 


1.0931 


1-3117 


I-5303 


1.7489 


1.9676 


o.3i45 


5 9 


0.2189 


0.4378 


0.6566 


0.8755 


1.0944 


I.3I33 


1-5322 


1. 7510 


1.9699 


0.3149 


60 



u 



114 


DISTANCES. 


13° 


oo 


1 


3 


3 


4 


5 


6 


7 


8 


9 


a 


0.9481 


1. 8961 


2.8442 


3-7923 


4.7403 


5.6884 


6.6365 


7.5845 


8.5320 


1.3641 


OI 


0.9479 


1.8959 


2.8438 


3.79I8 


4-7397 


5.6876 


6.6356 


7.5835 


8.5315 


1.3040 


02 


0.9478 


1.8956 


2.8434 


3.7912 


4-7391 


5.6869 


6.6347 


7-5825 


8.5303 


1-3639 


03 


0.9477 


1.8954 


2.8431 


3.7907 


4-7384 


5.6861 


6.6338 


7-58I5 


8.5292 


1.3638 


04 


0.9476 


1.8951 


2.8427 


3.7902 


4-7378 


5.6853 


6.6329 


7.5804 


8.5280 


1-3637 


°5 


0.9474 


1.8949 


2.8423 


3.7897 


4-7371 


5.6846 


6.6320 


7-5794 


8.5269 


1.3636 


06 


o.9473 


1.8946 


2.8419 


3.7892 


4.7365 


5.6838 


6.631 1 


75784 


8.5257 


1-3635 


07 


0.9472 


1.8943 


2.8415 


3-7887 


4-7359 


5.6830 


6.6302 


7-5774 


8.5245 


1-3634 


08 


0.9470 


1. 8041 


2. 841 1 


3.7882 


4-7352 


5.6823 


6.6293 


7.5763 


8.5234 


1.3634 


09 


0.9469 


1.8938 


2.8407 


3-7877 


4.7346 


5.6815 


6.6284 


7-5753 


8.5222 


1-3633 


10 


0.9468 


1.8936 


2.8404 


3-787I 


4-7339 


5.6807 


6.6275 


7-5743 


8.5211 


1.3632 


11 


0.9467 


1.8933 


2.8400 


3.7866 


4-7333 


5-6799 


6.6266 


7-5733 


8.5199 


1. 3631 


12 


0.9465 


1.8931 


2.8396 


3.7861 


4.7326 


5.6792 


6.6257 


7.5722 


8.5188 


1-3630 


13 


0.9464 


1.8928 


2.8392 


3-7856 


4.7320 


5.6784 


6.6248 


7-5712 


8.5176 


1.3629 


14 


0.9463 


1.8925 


2.8388 


3-785I 


4-73I3 


5-6776 


6.6239 


7-5702 


8.5164 


1.3628 


15 


0.9461 


1.8923 


2.8384 


3.7846 


4.7307 


5-6708 


6.6230 


7-5691 


8.5153 


1.3627 


16 


0.9460 


1.8920 


2.8380 


3.7840 


4.7300 


5.6761 


6.6221 


7-5681 


8.5141 


1.3626 


17 


0.9459 


1. 8918 


2.8376 


3-7835 


4.7294 


5.6753 


6.6212 


7-5670 


8.5129 


1-3625 


18 


0.9458 


1.8915 


2.8373 


3-7830 


4.7288 


5.6745 


6.6203 


7.5660 


8.5118 


1.3624 


19 


0.9456 


1. 8912 


2.8369 


3-7825 


4.7281 


5.6737 


6.6193 


7-5650 


8.5106 


1.3623 


20 


o-9455 


1. 8910 


2.8365 


3.7820 


4.7275 


5.6729 


6.6184 


7.5639 


8.5094 


1.3622 


21 


0.9454 


1.8907 


2.8361 


3-78I4 


4.7268 


5.6722 


6.6175 


7.5629 


8.5082 


1. 3621 


22 


0.9452 


1.8905 


2.8357 


3.7809 


4. 7261 


5-6714 


6.6166 


7.5618 


8.5071 


1.3620 


23 


0.9451 


1.8902 


2.8353 


3-7804 


4.7255 


5.6706 


6.6157 


7.5608 


8.5059 


1.3619 


24 


0.9450 


1.8899 


2.8349 


3-7799 


4.7248 


5.6698 


6.6148 


7-5597 


8.5047 


1.3618 


25 


0.9448 


1.8897 


2.8345 


3-7793 


4.7242 


5.6690 


6.6139 


7.5587 


8.5035 


1.3618 


26 


0-9447 


1.8894 


2.8341 


3-7788 


4.7235 


5.6682 


6.6129 


7.5576 


8.5023 


1-3617 


27 


0.9446 


1. 8891 


2.8337 


3-7783 


4.7229 


5.6674 


6.6120 


7-5566 


8.5012 


1.3616 


28 


0.9444 


1.8889 


2.8333 


3.7778 


4. 7222 


5.6667 


6.61 1 1 


7-5555 


8.5000 


1.3615 


29 


0.9443 


1.8886 


2.8329 


3-7772 


4.7216 


5-6659 


6.6102 


7-5545 


8.4988 


1-3614 


30 


0.9442 


1.8884 


2.8325 


3-7767 


4.7209 


5.6651 


6.6093 


7-5534 


8.4976 


1.3613 


3i 


0.9440 


1.8881 


2.8321 


3.7762 


4.7202 


5-6643 


6.6083 


7-5524 


8.4964 


1. 3612 


32 


0.9439 


1.8878 


2.8317 


3-7757 


4.7196 


5.6635 


6.6074 


7-55^3 


8.4952 


1.3611 


33 


0.9438 


1.8876 


2.8313 


3.7751 


4.7189 


5.6627 


6.6065 


7-5503 


8.4940 


1. 3610 


34 


0.9436 


1.8873 


2.8309 


3-7746 


4.7182 


5.6619 


6.6055 


7.5492 


S.4928 


1.3609 


35 


o.9435 


1.8870 


2.8306 


3-7741 


4.7176 


5.6611 


6.6046 


7-548i 


S.4917 


1.3608 


36 


o.9434 


1.8868 


2.8302 


3-7735 


4.7169 


5.6603 


6.6037 


7-5471 


8.4905 


1.3607 


37 


0.9433 


1.8865 


2.8298 


3-7730 


4-7 l6 3 


5-6595 


6.6028 


7-5460 


S.4893 


1.3606 


38 


0.9431 


1.8862 


2.8294 


3.7725 


4- 7^6 


5-6587 


6.601S 


7-5449 


S.4881 


1-3605 


39 


0.9430 


1.8860 


2.8290 


3.77I9 


4.7149 


5.6579 


6.6009 


7-5439 


8.4S69 


1.3604 


4o 


0.9429 


1.8857 


2.8286 


3.7714 


4-7I43 


5.6571 


6.6000 


7.5428 


8.4857 


1.3603 


4i 


0.9427 


1.8854 


2.8282 


3.7709 


4-7136 


5-6563 


6.5990 


7.54i8 


8.4845 


1.3602 


42 


0.9426 


1.8852 


2.8278 


3-7703 


4.7129 


5.6555 


6.5981 


7.5407 


8.4833 


1.3602 


43 


0.9425 


1.8849 


2.8274 


3.7698 


4-7123 


5-6547 


6.5972 


7-5396 


S.4S21 1 


1. 3601 


44 


0.9423 


1.S846 


2.8270 


3-7693 


4-7116 


5-6539 


6.5962 


7-5385 


S.4S09 


1.3600 


45 


0.9422 


1.8844 


2.8265 


3.76S7 


4.7109 


5-653I 


6-5953 


7-5375 


8.4796 ; 


1-3599 


46 


0.9420 


1. 8841 


2.8261 


3.7682 


4.7102 


5.6523 


6-5943 


7-5364 


S.47S4 


I.359S 


47 


0.9419 


1.8838 


2.8257 


3-7677 


4.7096 


5-6515 


6-5934 


7-5353 


8.4772 


1-3597 


48 


0.9418 


1.8836 


2.8253 


3-7671 


4.70S9 


5-6507 


0.5925 


7-5342 


s.4700 


I-359 6 


49 


0.9416 


1.8833 


2.8249 


3.7666 


4.70S2 


5-6499 


6-59*5 


7-5332 


8.4748 


1-3595 


5o 


c.9415 


1.S830 


2.8245 


3.7660 


4.7076 


5.6491 


6.5906 


7-5321 


8.4736 


1-3594 


5i 


0.9414 


1.8828 


2.S241 


3-7655 


4.7069 


5.6483 


6.5S96 


7-53IO 


8.4724 


1-3593 


52 


0.9412 


1.8825 


2.8237 


3-7650 


4.7062 


5.6474 


6.5SS7 


7-5299 


s.4712 


I-359 2 


53 


0.941 1 


1.8822 


2.8233 


3-7644 


4.7055 


5.6466 


6.5877 


7-52SS 


8.4699 


1-359^ 


55 


0.9410 


1. 8819 


2.8229 


3-7639 


4-7048 


5.6458 


6.5S6S 


7-5278 


8.46S7 


I.3S90 


55 


0.9408 


1.8817 


2.8225 


3-7633 


4.7042 


5.6450 


6.5S5S 


7-5267 


8.4675 ; 


1.3589 


56 


0.9407 


1.8S14 


2.8221 


3-7628 


4.7035 


5.6442 


6.5S49 


7-5256 


s.4663 


1.3588 


57 


0.9406 


1.8811 


2.8217 


3-7623 


4.7028 


5.6434 


6.5839 7.5245 


s.4651 


I-35S7 


58 


0.9404 


1.8809 


2.8213 


3-7617 


4.7021 


5.6426 


6.5830 : 7-5234 


S.463S 




59 


0.9403 


1.8806 


2.S209 


3.7612 


4-7015 


5.6418 


6.5S20 j 7.5223 


s.4026 


r.3585 


6o 


0.9402 


1.8803 


2.8205 


3.7606 


4.700S 


5.6409 


6.5S11 7.5212 


s.4014 


1.3584 



13° 


HEIGHTS. 




! 
115 


1 


3 


3 


4 

0-8755 


5 


6 


7 


8 


9 


b 


1 
00 


0.2189 


0.4378 


0.6566 


I.0944 


I.3I33 


1.5322 


1. 7510 


1.9699 


0.3149 


0.2IQI 


0-4383 


0.6574 


0.8766 


I.0957 


1.3148 


1-5340 


i- 7531 


1.9723 


0.3153 


01 


0.2194 


0.4388 


0.6582 


0.8776 


1.0970 


1. 3164 


1.5358 


1-7552 


1.9746 


o.3i57 


J02 


0.2197 


o.4393 


0.6550 


c.8786 


1.0583 


1.3180 


1-5376 


1-7573 


1.9769 


0.3161 


!o 3 


0.2199 


0.4398 


0.6598 


0.8797 


1.0996 


1.3*95 


1-5394 


1-7594 


x -9793 


0.3165 


'04 


0.2202 


0.4404 


0.6605 


0.8807 


1. 1009 


1.3211 


I-54I3 


1. 7614 


1. 9816 


0.3169 


05 


0.2204 


0.4409 


0.6613 


0.8818 


1. 1022 


1.3226 


1. 543i 


1-7635 


1.9840 


o-3!73 


06 


0.2207 


0.4414 


0.6621 


0.8828 


1-1035 


1.3242 


1-5449 


1.7656 


1.9863 


0.3177 


07 


0.2210 


0.4419 


0.6629 


0.8838 


1. 1048 


1.3258 


1-5467 


1.7677 


1.9886 


0.3181 


jo8 


0.2212 


0.4425 


0.6637 


0.8849 


1.1061 


1-3274 


1.5486 


1.7698 


1.9911 


0.3185 


09 


0.2215 


0.4430 


0.6645 


0.8860 


1. 1074 


1.3289 


1-5504 


1.7719 


1-9934 


0.3189 


10 


0.2217 


0-4435 


0.6652 


0.8870 


1.1087 


I.3305 


1.5522 


1.7740 


1-9957 


o.3 J 93 


11 


0.2220 


0.4440 


0.6660 


0.8880 


T.IIOO 


I.332I 


I-554I 


1. 7761 


1. 9981 


0.3197 


|l2 


0.2223 


0.4445 


0.6668 


0.8891 


I.III3 


1.3336 


1-5559 


1.7782 


2.0004 


0.3201 


13 


0.2225 


0.4451 


0.6676 


0.8901 


1. 1 126 


1-3352 


1-5577 


1.7802 


2.0028 


0.3205 


14 


0.2228 


0.4456 


0.6684 


0.8912 


1. 1 139 


*-33 6 7 


1-5595 


1.7823 


2.0051 


0.3209 


IS 


o. 2230 


0.4461 


0.6691 


0.8922 


I.II52 


1.3383 


J -56i3 


1.7844 


2.0074 


0.3213 


l6 


O.2233 


0.4466 


0.6699 


0.8932 


1. 1 165 


L3399 


1-5632 


1.7865 


2.0098 


0.3217 


17 


O.2236 


0.4471 


0.6707 


0.8943 


I.II78 


I-34I4 


1.5650 


1.7886 


2.0121 


0.3221 


18 


O.2238 


o.4477 


0.6715 


c.8953 


I.II91 


I-3430 


1.5668 


1.7906 


2.0145 


0.3225 


19 


O.224I 


0.4482 


0.6723 


0.8964 


I. 1204 


!-3445 


1.5686 


1.7927 


2.0168 


0.3229 


20 


O.2243 


0.4487 


0.6730 


0.8974 


I.I217 


1.3460 


1-5704 


1-7947 


2.0191 


0.3232 


21 


O.2246 


0.4492 


0.6738 


0.8984 


1. 1230 


1.3476 


1.5722 


1.7968 


2.0214 


0.3236 


22 


O.2249 


0.4497 


0.6746 


0.8994 


1. 1243 


1.3492 


1-5740 


1.7989 


2.0237 


0.3240 


23 


O.225I 


0.4502 


0.6754 


0.9005 


1. 1256 


J-3507 


1.5758 


1. 8010 


2.0261 


0.3244 


24 


O.2254 


0.4508 


0.6761 


0.9015 


1. 1269 


1-3523 


1-5777 


1.8030 


2.0284 


0.3248 


25 


O.2256 


o.45i3 


0.6769 


0.9026 


1. 1282 


1.3538 


1-5795 


1.8051 


2.0308 


0.3252 


26 


O.2259 


0.4518 


0.6777 


0.9036 


I. I295 


1-3554 


1.5813 


1.8072 


2.0331 


0.3256 


27 


0. 2262 


0.4523 


0.6785 


0.9046 


1. 1308 


1-3570 


1.5831 


1.8093 


2.0354 


0.3260 


I28 


O.2264 


0.4528 


0.6793 


0.9057 


I.1321 


I-3585 


1.5849 


1.8114 


2.0378 


0.3264 


29 


O.2267 


0-4534 


0.6800 


0.5067 


I- 1334 


1. 3601 


1.5868 


1.8134 


2.0401 


0.3268 


30 


O.2269 


o.4539 


0.6808 


0.5078 


I- 1347 


1.3616 


1.5886 


i.8i55 


2.0424 


0.3272 


31 


O.2272 


0.4544 


0.6816 


0.9088 


I.I360 


1. 3631 


1.5904 


1.8175 


2.0447 


0.3276 


32 


O.2275 


o.4549 


0.6824 


0.9098 


I- 1373 


1-3647 


1.5922 


1. 8196 


2.0471 


0.3280 


33 


O.2277 


o.4554 


0.6831 


0.9108 


I.I38S 


1.3663 


1.5940 


1. 8217 


2.0494 


0.3284 


34 


O.2280 


0.4559 


0.6839 


0.9119 


1. 1398 


1.3678 


1-5958 


1.8238 


2.0517 


0.3288 


|35 


0. 2282 


0.4565 


0.6847 


0.9129 


I.14H 


1.3694 


1.5976 


1.8258 


2.0541 


0.3292 


36 


O.2285 


0.4570 


0.6855 


0.9140 


1. 1424 


1.3709 


1-5994 


1.8279 


2.0564 


0.3296 


37 


O.2287 


0.4575 


0.6862 


0.9150 


T- 1437 


1-3725 


1. 6012 


1.8300 


2.0587 


0.3300 


38 


O.229O 


0.4580 


0.6870 


0.9160 


I. I45O 


1-3740 


1.6030 


1.8320 


2.0610 


0.3304 


39 


O.2293 


0.4585 


0.6878 


0.9170 


1. 1463 


1.3756 


1.6048 


1.8341 


2.0633 


0.3308 


40 


O.2295 


0.4590 


0.6886 


0.9181 


1. 1476 


I-377I 


1.6066 


1.8361 


2.0657 


0.3312 


4i 


O.2298 


0.4596 


0.6993 


0.9191 


1. 1489 


I-3787 


1.6085 


1.8382 


2.0680 


0.3316 


42 


O.23OO 


0.4601 


0.6901 


0.9202 


I. I502 


1.3802 


1. 6103 


1.8403 


2.0703 


0.3320 


J43 


O.23O3 


0.4606 


0.6909 


0.9212 


I-I5I5 


1.3817 


1.6121 


1.8423 


2.0726 


0.3324 


44 


O.2306 


0.461 1 


0.6917 


0.9222 


I. 1528 


1.3833 


1.6139 


1.8444 


2.0750 


0.3328 


,45 


O.2308 


0.4616 


0.6924 


0.9232 


I.I54I 


1.3849 


i-6i57 


1.8465 


2.0773 


0.3331 


46 


O.23I I 


0.4621 


0.6932 


0.9243 


I- 1554 


1.3864 


1.6175 


1.8486 


2.0796 


o.3335 


47 


O.2313 


0.4626 


0.6940 


o.9253 


1. 1566 


I-3879 


1-6193 


1.8506 


2.0819 


0-3339 


48 


O.2316 


0.4632 


0.6947 


0.9263 


I- 1579 


1.3895 


1.6211 


1.8527 


2.0842 


o.3343 


49 


O.2318 


0.4637 


0.6955 


0.9274 


I. 1592 


1.3910 


1.6229 


1-8547 


2.0866 


o.3347 


50 


O.232I 


0.4642 


0.6963 


0.9284 


1. 1605 


1.3926 


1.6247 


1.8568 


2.0889 


o.335i 


i 51 


O.2324 


0.4647 


0.6971 


0.9294 


I.l6l8 


1 -3941 


1.6265 


1.8588 


2.0912 


o.3355 


52 


O.2326 


0.4652 


0.6978 


0.9304 


1. 163O 


1-3957 


1.6283 


1.8609 


2.0935 


o.3359 


!53 


O.2329 


0.4657 


0.6986 


o.93i5 


1. 1643 


1.3972 


1. 6301 


1.8630 


2.0958 


0.3363 


54 


O.233I 


0.4662 


0.6994 


0.9325 


1. 1656 


I-3987 


1. 6319 


1.8650 


2.0981 


0.3367 


! 55 


O.2334 


0.4668 


0.7001 


o.9335 


1. 1669 


1.4003 


1-6337 


1.8670 


2. 1004 


0-337 1 


56 


O.2336 


0.4673 


0.7009 


0.9346 


1. 1682 


1. 4018 


1.6355 


1.8691 


2. 1028 


o.3375 


J57 


O.2339 


0.4678 


0.7017 


0-935 6 


1. 1695 


I-4033 


1.6373 


1.8711 


2. 1050 


o.3379 


58 


O.2342 


0.4683 


0.7025 


0.9366 


1. 1708 


1.4049 


1.6391 


1.8732 


2.1074 


0.3383 


59 


O.2344 


0.4688 


0. 7032 


0.9376 


1. 1720 


1.4065 


1.6409 


1.8753 


2.1097 


0.3387 


60 



116 DISTANCES. 


14° 


/ 
oo 


1 


3 


3 


4 


5 


6 


7 


8 


9 


a 


0.9402 


1.8803 


2.8205 


3.7606 


4. 7008 


5.6409 


6.5811 


7.5212 


8.4614 


1-3584 


OI 


0.9400 


1.8800 


2.8201 


3.7601 


4.7001 


5.6401 


6.5801 


7.5202 


8.4602 


1.3583 


02 


0.9399 


1.8798 


2.8196 


3-7595 


4.6994 


5-6393 


6.5792 7-5191 


8.4589 


1.3582 


03 


o.9397 


1.8795 


2.8192 


3-7590 


4.6987 


5-6385 


6.5782 7.5180 


8-4577 


i.358i 


04 


0.9396 


1.8792 


2.8188 


3.7584 


4.6980 


5-6376 


6-5773 


7.5169 


8.4565 


1.3580 


°5 


0-9395 


1.8789 


2.8184 


3-7579 


4.6974 


5.6368 


6.5763 


7.5I58 


8.4552 


1-3579 


o5| 


o.9393 


1.8787 


2.8180 


3-7573 


4.6967 


5-6360 


6-5753 


7.5I47 


8.4540 


1.3578 


07I 


o-9392 


1.8784 


2.8176 


3.7568 


4.6960 


5.6352 


6-5744 


7-5I36 


8.4528 


1-3577 


oS 


0.9391 


1. 8781 


2.8172 


3-7562 


4-6953 


5.6344 


6-5734 


7-5125 


8.4515 


1-3576 


0; 


0.9389 


1.8778 


2.8168 


3-"7557 


4.6946 


5.6335 


6.5725 


7-5II4 


8.4503 


1-3575 


10 


0.0388 


1.8776 


2.8164 


3.7551 


4.6939 


5-6327 


6.5715 


7.5103 


8.4491 


1-3574 


11 


0.9386 


1.8773 


2.8159 


3.7546 


4.6932 


5-6319 


6.5705 


7.5092 


8.4478 


1-3573 


12 


0.9385 


1.8770 


2.8155 


3.7540 


4.6925 


5-6310 


6.5696 


7.5081 1 8.4466 


1.3572 


J 3 


0.93S4 


1.8767 


2.8151 


3-7535 


4.6918 


5.6302 


6.5686 


7.5070 ! 8.4453 


I-357* 


14 


0.9382 


1.8765 


2.8147 


3.7529 


4.6912 


5.6294 


6.5676 


7.5058 8.4441 


' I-3570 


IS 


0.9381 


1.8762 


2.8143 


3-7524 


4.6905 


5.6286 


6.5666 


7.5047 8.4428 


1-3569 


16 


0.9380 


1.8759 


2.8139 


3-7518 


4.6898 


5-6277 


6.5657 


7-5036 


8.4416 


1.3568 


17 


0.9378 


1.8756 


2.8134 


3-7513 


4.6891 


5.6269 


6.5647 


7-5025 


8.4403 


1-3567 


18 


o.9377 


1.8754 


2.8130 


3-7507 


4.6884 


5.6261 


6.5637 


7-5014 


8.4391 


1.3566 


J 9 


o.9375 


1.8751 


2.8126 


3- 750i 


4.6877 


5-6252 


6.5628 


7.5003 1 8.4378 


1-3565 


20 


o.9374 


1.8748 


2.8122 


3-7496 


4.6870 


5-6244 


6.5618 


7.4992 


8.4366 


1-3564 


21 


o.9373 


1.8745 


2.8118 


3-7490 


4.6863 


5-6235 


6.560S 


7.4981 


8-4353 


1.3563 


22 


o.937i 


1.S742 


2.8114 


3-74S5 


4.6856 


5.6227 


6.5598 


7.4969 


8.4341 


1-3562 


23 


0.9370 


1.8740 


2.8109 


3-7479 


1 4.6849 


5.6219 


6.5588 


7-4958 


8.4328 


i-356i 


24 


0.9368 


1.8737 


2.8105 


3-7474 


4.6842 


5.6210 


6-5579 


7-4947 


8.4315 


1.3560 


25 


0.9367 


1.8734 


2.8101 


3.7468 


4.6835 


5.6202 


6.5569 


7.4936 


8.4303 


1-3559 


261 


0.9366 


1.8731 


2.8097 


3.7462 


4.6828 


5.6i93 


6-5559 


7.4925 


8.4290 


1.3558 


271 


0.9364 


1.8728 


2.8093 


3-7457 


4.6821 


5.6185 


6-5549 


7.49I3 


8.4278 


1-3557 


28 


o.93 6 3 


1.8726 


2.80S8 


3-7451 


4.6814 


5.6i77 


6-5539 


7.4902 


8.4265 


1.3556 


29 


0.9361 


1.8723 


2.8084 


3-7445 


4.6807 


5.6168 


6.5530 


7.4891 


8.4252 


1-3555 


30 


0.9360 


1.8720 


2.8080 


3-7440 


4.6800 


5.6160 


6.5520 


7.4880 


8.4240 


1-3554 


3i 


o.9359 


1.8717 


2.8076 


3-7434 


4-6793 


5-6151 


6.5510 


7.4868 


8.4227 


1-3553 


32 


0-9357 


1. 8714 


2.8071 


3-7429 


4.6786 


5-6143 


6.55^0 


7.4857 


8.4214 . 


1-3552 


33 


0.9356 


1.8711 


2.8067 


3-7423 


4.6779 


5-6i34 


6.5490 


7.4846 


8.4202 


I.355I 


34 


o.9354 


1.8709 


2.8063 


3-74I7 


4.6772 


5.6126 


6.54S0 


7.4834 


8.4189 


I.3550 


35 


o.9353 


1.8706 


2.8059 


3-7412 


4.6764 


5-6ii7 


6.5470 


7-4S23 


8.4176 


1-3549 


36 


o.935i 


1.8703 


2.8054 


3.7406 


4-6757 


5.6109 


6.5460 


7.4S12 


S.4163 


I.354S 


37 


0.9350 


1.8700 


2.8050 


3.7400 


4.6750 


5.6100 


6.5450 


7.4801 


8.4151 


1-3547 


38 


0-9349 


1.8697 


2.S046 


3-7395 


4-6743 


5.6092 


6.5441 


7-4789 


8.413S 


I-3546 


39 


0-9347 


1.8694 


2.8042 


3-73S9 


4-6736 


5.60S3 


6.5431 


7.4778 


8.4125 


1-3545 


40 


0.9346 


1.8692 


2.8037 


3.7383 


4.6729 


5.6075 


6.5421 


7-4767 


S.4112 


1.3544 


4i 


0.9344 


1.8689 


2.8033 


3.7378 


4.6722 


5.6066 


6. 541 1 


7-4755 


8.4100 


1-3543 


42 


o.9343 


1.86S6 


2.8029 


3-7372 


4-67I5 


5.6058 


6.5401 


7-4744 


8.40S7 


1-3542 


43 


o.9342 


1.8683 


2.S025 


3-7366 


4.6708 


5.6049 


6.5391 


7-4732 


8.4074 


I-354I 


44 


0.9340 


1.86S0 


2.8020 


3-736o 


4.6701 


5.6041 


6.53S1 


7-4721 


8.4061 


1 -3540 


45 


o.9339 


1.8677 


2.8016 


3-7355 


4.6693 


5-6032 


6.5371 


7-4709 


8.4048 


1-3539 


46 


o-9337 


1.8674 


2.8012 


3-7349 


4.66S6 


5-6023 


6.5361 


7.469S 


S.4035 


I.353S 


47 


o.933 6 


1.8672 


2.8007 


3-7343 


4.6679 


5.6015 


6.5351 


7.46S6 


8.4022 


1-3537 


48 


o.9334 


1.8669 


2.8003 


3-7338 


4.6672 


5.6006 


6.5341 


7-4675 


8.4009 : 


I-3536 


49 


0-9333 


1.8666 


2-7999 


3-7332 


4.6665 


5-5998 


6.5331 


7.4664 


S.3997 


1.3535 


50 


o.9332 


1.8663 


2-7995 


3-7326 


4.665S 


5.5989 


6.5321 


7-4652 


S.39S4 


1-3534 


5i 


0.9330 


1.8660 


2.7990 


3-7320 


4.6650 


5.598o 


6.53" 


7.4641 


S.397I 


1-3533 


52 


0.9329 


1.8657 


2.79S6 


3-73I5 


4.6643 


5-5972 


6.5300 


7.4629 


8.3958 


1 -353 1 


53 


0.9327 


1.S654 


2.7982 


3-7309 


4.6636 


5.5963 


6.5290 


7.461S 


S.3945 


I-3530 


54 


0.9326 


1.8651 


2.7977 


3-7303 


4.6629 


5-5954 


6.52S0 


7.4606 


8.3932 


1-3529 


55 


0.9324 


1.8649 


2-7973 


3.7297 


4.6621 


5.5946 


6.5270 


7-4594 


S.39I9 


1.3528 


56 


0.9323 


1.8646 


2.7969 


3-7291 


4.6614 


5-5937 


6.5260 


7.4583 


S.3906 




57 


0.9321 


1.8643 


2.7964 


3-72S6 


4.6607 


5o928 


6.5250 


7-4571 


S.3893 


1.3526 


58 


0.9320 


1.8640 


2.7960 


3.72S0 


4.6600 


5-5920 


6.5240 


7-4560 




1 ^523 


59 


0.9319 


1.8637 


2.7956 


3.7274 1 4.6593 


5-59" 


6.5230 


7.454S 


8.3867 


1-35*4 


60 


0.9317 


1.S634 


2.7951 3-7268)14.6585 


5-5902 


6.5219 


7-4537 


S.3S54 


1.3523 



14° 


HEIGHTS. 


117 


1 


3 


3 


4 


5 


6 


7 


8 


9 


b 


' 


o.2344 


0.4688 


0.7032 


O.9376 


1. 1720 


1.4065 


1.6409 


1.8753 


2. 1097 


0.3387 


00 


0.2347 


0.4693 


0. 7040 


0.9386 


1. 1733 


1.4080 


1.6426 


1.8773 


2. 1 120 


0-339 1 


joi 


0.2349 


0.4698 


0. 7048 


o.9397 


1. 1746 


1.4095 


1.6444 


1.8794 


2. 1 143 


0-3395 


J02 


.0.2352 


0.4704 


0.7055 


0.9407 


Li 759 


1.4111 


1.6462 


1. 8814 


2. 1 166 


0.3399 


l° 3 


Q.2354 


0.4709 


0.7063 


0.9417 


1. 1772 


1. 4126 


1.6480 


1.8834 


2.1189 


0.3403 


04 


Q.2357 


0.4714 


0.7071 


0.9428 


1. 1 785 


1.4141 


1.6498 


1.8855 


2.1212 


0.3407 


°5 


o-2359 


0.4719 


0.7078 


0.9438 


I- 1 797 


1. 4156 


1. 6516 


1.8875 


2.1235 


0.341 1 


06 


0.2362 


0.4724 


0. 7086 


0.9448 


1. 1810 


1. 4172 


I-6534 


1.8896 


2.1258 


0.3414 


07 


0.2365 


0.4729 


0.7094 


0.9458 


1. 1823 


1. 4188 


1.6552 


1.8917 


2.1281 


0.3418 


08 


0.2367 


o.4734 


0.7101 


0.9468 


1. 1836 


1.4203 


1.6570 


1.8937 


2.1304 


0.3422 


09 


0.2370 


o.4739 


0.7109 


0.9479 


1. 1848 


1. 4218 


1.6588 


1.8958 


2.1327 


0.3426 


10 


0.2372 


0.4744 


0.7117 


0.9489 


1.1861 


L4233 


1.6606 


1.8978 


2.1350 


o-343° 


11 


0-2375 


0.4750 


0.7124 


o.9499 


1. 1874 


1.4249 


1.6624 


1.8998 


2.1373 


0-3434 


12 


0.2377! 


o.4755 


0.7132 


0.9509 


1. 1887 


1.4264 


1. 6641 


1. 9018 


2. 1396 


o.3438 


13 


0.2380, 


0.4760 


0.7140 


0.9520 


1. 1899 


1.4279 


1.6659 


1.9039 


2.1419 


0.3442 


14 


0.2382 


0.4765 


0.7147 


0.9530 


1.1912 


1.4295 


1.6677 


1.9060 


2.1442 


0.3446 


15 


0.2385 


0.4770 


o.7i55 


0.9540 


1. 1925 


1. 4310 


1.6695 


1.9080 


2.1465 


o.345o 


16 


0.2388 


0-4775 


0.7163 


0.9550 


1.1938 


1.4326 


1-6713 


1.9101 


2.1488 


C-3454 


17 


0.2390 


0.4780 


0.7170 


0.9560 


1.1951 


1 -4341 


1-6731 


1.9121 


2.1511 


o.3458 


18 


o.2393 


0.4785 


0.7178 


o.957i 


1. 1963 


I.4356 


1.6749 


1. 9142 


2.1534 


0.3462 


19 


Q.2395 


0.4790 


0.7186 


c.9581 


1. 1976 


1 -437 1 


1.6767 


1. 9162 


2.1557 


0.3466 


20 


0.2398 


0.4796 


0.7193 


0.9591 


1. 1989 


1.4387 


1.6785 


1. 9182 


2.1580 


0.3470 


21 


0.2400 


0.4801 


0.7201 


0.9601 


1.2002 


1.4402 


1.6802 


1.9202 


2.1603 


o.3474 


22 


0.2403 


0.4806 


0.7209 


0.961 1 


1. 2014 


I.44I7 


1.6820 


1.9223 


2.1626 


0.3478 


23 


0.2405 


0.481 1 


0.7216 


0.9622 


1.2027 


1.4432 


1.6838 


1.9243 


2.1648 


0.3482 


24 


0.2408 


0.4816 


0. 7224 


0.9632 


1.2040 


1.4448 


1.6856 


1.9264 


2.1671 


0.3485 


25 


0.241 1 


0.4821 


0.7232 


0.9642 


1.2053 


1.4463 


1.6874 


1.9284 


2. 1694 


0.3489 


26 


0.2413 


0.4826 


0.7239 


0.9652 . 


1.2065 


1.4478 


1. 6891 


1.9304 


2.1717 


o.3493 


27 


0.2416 


0.4831 


0.7247 


0.9662 


1.2078 


1.4494 


1.6909 


I-9325 


2.1740 


o.3497 


,28 


0.2418 


0.4836 


0.7254 


0.9672 


1. 2091 


1.4509 


1.6927 


1-9345 


2.1763 | 


0.3501 


29 


0.2421 


0.4841 


0. 7262 


0.9683 


1. 2103 


1.4524 


1.6945 


1.9366 


2.1786 


0.3505 


30 


0.2423 


0.4846 


0. 7270 


0.9693 


1.2116 


1-4539 


1.6962 


1.9386 


2.1809 


0.3509 


3 1 


0.2425 


0.4851 


0.7277 


0.9703 


1. 2129 


1-4554 


1.6980 


1.9406 


2.1831 | 


o.35i3 


32 


0.2428 


0.4857 


0.7285 


o.97i3 


1. 2 141 


I-4570 


1.6998 


1.9426 


2.1855 j 


o.35i7 


\33 


0.2431 


0.4862 


0. 7292 


0.9723 


1.2154 


1.4585 


1. 7016 


1.9446 


2.1877 


0.3521 


34 


0.2433 


0.4867 


0.7299 


o.9733 


1. 2166 


1.4600 


1-7033 


1.9466 


2.1900 


0.3525 


J35 


0.2436 


0.4872 


0.7307 


o.9743 


1.2179 


1.4615 


1. 7051 


1.9487 


2.1923 


0.3529 j 


36 


0.2438 


0.4877 


o.73i5 


o.9754 


1. 2192 


1.4630 


1.7069 


1-9507 


2.1946 


0.3533 j 


37 


0.2441 


0.4882 


0.7323 


0.9764 


1.2205 


1.4646 


1.7087 


1.9528 


2.1969 


0.3537 


38 


0.2443 


0.4847 


0.7330 


o.9774 


1. 2217 


1. 4661 


1. 7104 


1.9548 


2. 1991 


0.3541 


39 


0.2446 


0.4892 


o.7338 


0.9784 


1.2230 


1.4676 


1. 7122 


1.9568 


2.2014 


0.3545 


40 


0.2449 


0.4897 


0.7346 


o.9794 


1.2243 


1.4691 


1. 7140 


1.9588 


2.2037 


0.3549 


4i 


0.2451 


0.4902 


o.7353 


0.9804 


1.2255 


1.4706 


I-7I57 


1.9608 


2.2059 


0.3553 


42 


0.2454 


0.4907 


0.7361 


0.9814 


1.2268 


1.4722 


I.7I75 


1.9629 


2.2082 


0.3556 


43 


0.2456 


0.4912 


0.7368 


0.9824 


1. 2281 


1-4737 


1. 7193 


1.9649 


2.2105 


0.3560 


44 


0.2459 


0.4917 


0.7376 


0.9835 


1.2294 


1.4752 


1.7211 


1.9670 


2.2128 


0.3564 


45 


0.2461 


0.4922 


0.7384 


0.9845 


1.2306 


1.4767 


1.7228 


1.9690 


2.2151 


0.3568 


46 


0.2464 


0.4927 


0-739 1 


0.9855 


1.2319 


1.4782 1.7246 


1.9710 


2.2173 


0.3572 


47 


0.2466 


0.4932 


o.7399 


0.9865 


1.2331 


1-4797 


1.7263 


I.9730 


2.2196 


0.3576 


48 


0.2469 


0.4938 


0.7406 


0.9875 


1.2344 


1. 4813 


1. 7281 


i-975o 


2.2219 


0.3580 


49 


0.2471 


0.4943 


0.7414 


c.9885 


1.2357 


1.4828 


1.7299 


1.9770 


2.2242 


0.3584 


5o 


0.2474 


0.4948 


0.7421 


0.9895 


1.2369 


1.4843 


1. 7317 


1.9790 


2.2264 


0.3588 


5i 


0.2476 


0-4953 


0.7429 


0.9905 


1.2382 


1.4858 


1-7334 


1. 9810 


2.2287 O.3592 | 


52 


0.2479 


o.4958 


0.7436 


0.9915 


1.2394 


1.4873 


1.7352 


1.9831 


2.2310 0.3596 


53 


0.2481 


0.4963 


0.7444 


0.9926 


1.2407 


1.4888 


1.7370 


1.9851 


2.2333 0.3600 


54 


0.2484 


0.4968 


0.7452 


0.9936 


1.2420 


1.4903 


1.7387 


1.9871 


2.2355 0.3604 ! 


55 


0.2486 


o.4973 


o.7459 


0.9946 


1.2432 


1.4918 


I-7405 


1.9891 


2.2378 ._ 0.3608 j 


56 


0.2489 


0.4978 


0.7467 


0.9956 


1.2445 


1-4933 


1.7422 


1.9911 


2.2400 


0.3612 


57 


0.2491 


0.4983 


c.7474 


0.9966 


L2457 


1.4949 


1.7440 


1.9932 


2.2423 


0.3616 


58 


0.2494 


0.4988 


0. 7482 


0.9976 


1.2470 


1.4964 


1-7458 


1.9952 


2.2446 : 


0.3620 


59 


0.2497 


0.4993 


0.7490 


0.9986 


1.2483 


1.4979 1.7476 


1.9972 


2.2469 ! 


0.3623 


60 



118 


DISTANCES. 


15° 


oo 


1 


3 


3 


4 


5 


6 


7 


8 


9 


a 


0.9317 


1.8634 


2.7951 


3.7268 


4.6585 


5-5902 


6.5219 


7.4537 


8.3854 


1-3523 


OI 


0.9316 


1.8631 


2-7947 


3.7262 


4.6578 


5.5894 


6.5209 


7.4525 


8.3840 


1-3522 


02 


0.9314 


1.8628 


2.7942 


3.7257 


4-6571 


5o885 


6.5199 


7.45I3 


8.3827 


I-352I 


03 


o-93i3 


1.8625 


2.7938 


3.725I 


4-6563 


5-5876 


6.5189 


7.4502 


8.3814 


1.3520 


04 


0.931 1 


1.8622 


2-7934 


3.7245 


; 4-6556 


5.5867 


6-5179 


7.4490 


8.3801 


I-35I9 


05 


0.9310 


1.8620 


2.7929 


3.7239 


! 4.6549 


5.5859 


6.5168 


7.4478 


8.3788 


i-35i8 


06 


0.9308 


1.8617 


2.7925 


3.7233 


4.6542 


5-5850 


6.5158 


7.4466 


8-3775 


I.35I7 


07 


0.9307 


1. 8614 


2.7921 


3-7227 


4.6534 


5-5841 


6.5148 


7-4455 


8.3762 


i.35i6 


08 


0.9305 


1.8611 


2.7916 


3.7222 


4.6527 


5.5832 


6.5138 


7-4443 


8-3749 


I-35I5 


09 


0.9304 


1.8608 


2.7912 


3..7216 


4.6520 


5-5824 


6.5128 


7-4432 


8.3736 


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10 


0.9302 


1.8605 


2.7907 


3.7210 


4.6512 


5.58I5 


6.5117 


7.4420 


8.3722 


I.35I3 


II 


0.9301 


1.8602 


2.7903 


3-7204 


4-6505 


5.5806 


6.5107 


7.4408 


8.3709 


I-35I2 


12 


0.9300 


1.8599 


2.7899 


3.7198, 


4.6498 


5-5797 


6.5097 


7-4396 


8.3696 


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13 


0.9298 


1.8596 


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3-7192 


4.6490 


5.5788 


6.5086 


7.4384 


8.3682 


I-35IO 


14 


0.9297 


1.8593 


2.7890 


3.7186! 


4.6483 


5-5779 


6.5076 


7-4373 


8.3669 


1.3509 


15 


0.9295 


1.8590 


2.7885 


3.7180 


4.6476 


5-5771 


6.5066 


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8.3656 


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16 


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1.8587 


2.7881 


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4.6468 


5.5762 


6.5055 


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17 


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3.7169 


4.6461 


5-5753 


6.5045 


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18 


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4-6453 


5-5744 


6.5035 


7.4325 


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1.8578 


2.7868 


3-7157 


4.6446 


5-5735 


6.5024 


7-43*4 


8.3603 


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20 


0.9288 


1.8575 


2.7863 


3-7151 


4.6439 


5o726 


6.5014 


7.4302 


8.3590 


1.3502 


21 


0.9286 


1.8572 


2.7859 


3-7145 


4-643I 


5.57I7 


6.5004 


7.4290 


8.3576 


I-3SOI 


22 


0.9285 


1.8570 


2.7854 


3-7139 


4.6424 


5.5709 


6.4993 


7.4278 


8.3563 


1.3500 


23 


0.9283 


1.8567 


2.7850 


3-7133 


4.6416 


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6.4983 


7.4266 


8-3549 


1-3499 


24 


0.9282 


1.8564 


2.7845 


3-7127 


4.6409 


5-5691 


6.4972 


7.4254 


8.3536 


I-349S 


25 


0.9280 


1.8561 


2.7841 


3-7121 


4.6431 


5.5682 


6.4962 


7.4242 


8.3523 


1-3497 


26 


0.9279 


1.8558 


2.7836 


3-7115 


4.6394 


5-5673 


6.4952 


7-4230 


8.3509 


I-349 6 


27 


0.9277 


1.8555 


2.7832 


3.7109 ' 


4.6387 


5-5664 


6.4941 


7.4218 


S.3496 


1-3495 


28 


0.9276 


1.8552 


2.7827 


3-7103 


4.6379 


5-5655 


6.4931 


7.4207 


8.3482 


1-3494 


29 


0.9274 


1.8549 


2.7823 


3-7097 


4.6372 


5-5646 


6.4920 


7.4I95 


8.3469 


1-3493 


30 


0.9273 


1.8546 


2.7819 


3-7091 


4.6364 


5.5637 


6.4910 


7-4183 


8.3456 


1-349* 


3i 


0.9271 


1.8543 


2.7814 


3-7085 


4.6357 


5.5628 


6.4899 


7-4171 


8.3442 


I-3490 


32; 


0.9270 


1.8540 


2.7809 


3-7079 


4.6349 


5o6i9 


6.4889 


7.4I59 


8.3428 


1-3489 


33 


0.9268 


1.8537 


2.7805 


3-7073 : 


4.6342 


5o6io 


6.4878 


7.4147 


8.3415 


1.3488 


34 


0.0267 


1.8534 


2.7S00 


3-7067 


4.6334 


5o6oi 


6.4S68 


7-4*35 


8.3401 - 


1.3487 


35 


0.9265 


1.8531 


2.7796 


3-7061 


4.6327 


5-5592 


6.4S57 


7-4x23 


8.3388 


1.3486 


36 : 


0.9264 


1.8528 


2.7791 


3-7055 


4.6319 


5-5583 


6.4847 


7.4111 


8. 3374 


1-3485 


37 


0.9262 


1.8525 


2.7787 


3-7049 


4.6312 


5-5574 


6.4836 


7.4098 


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1.34S4 


38 


0.9261 


1.8522 


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4.6304 


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2.7778 


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4.6297 


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6.4815 


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6.4835 


7.4062 


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0.9256 


1.8513 


2.7769 


3.7025 


4.6281 


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6.4794 


7.4050 


8.3307 


1-3479 


42 


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1. 8510 


2.7764 


3-7019 


4.6274 


5o529 


6.47S3 


7.403S 


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1.3478 


43 


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1.8506 


2.7760 


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4.6266 


5.55I9 


6-4773 


7.4026 


8. 32 79 


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2-7755 


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4.6259 


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6.4762 


7.4014 


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45 


0.9250 


1.8500 


2.7751 


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6.4751 


7.4002 


8.3252 


1-3475 


46 


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2.7746 


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6.4741 


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6.4730 


7-3977 


8.3225 


1-3473 


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0.9246 


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3.69S3 


4.622S 


5-5474 


6.4720 


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1-3472 


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1.8488 


2.7732 


3-6977 


4.6221 


5-5465 


6.4709 


7-3953 


8.3197 


I-3470 


50 


0.9243 


1.8485 


2.772S 


3.0970 


4.6213 


5.5456 


6.4698 


7-3941 


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1.3469 


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1.8482 


2.7723 


3.6964 


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7-3929 


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52 


0.9240 


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2.7719 


3.695s 


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53 


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2.7714 


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1.3466 


54 


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15° 




HEIGHTS. 


119 


1 


3 


3 


4 


5 


6 


7 


8 


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0.4993 


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0.3623 


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0.4998 


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1.2495 


1.4994 


1-7493 


1.9992 


2.2491 


0.3627 


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1.5009 


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0.3631 


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1. 0016 


1.2520 


1.5024 


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2.0093 


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1.0066 


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09 


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1.0086 


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2.5130 


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14 


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15 


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16 


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17 


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19 


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20 


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24 


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1.0306 


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32 


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1.0316 


1.2895 


1-5474 


1.8053 


2.0632 


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33 


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1.2908 


1.5489 


1. 8071 


2.0652 


2.3234 


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34 


0.2584 


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0.7752 


1.0336 


1.2920 


1-5504 


1.8088 


2.0672 


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0.3761 


35 


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0.7760 


1.0346 


1-2933 


1. 55i9 


1. 8106 


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2.3279 


0.3765 


36 


0.2589 


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1.0356 


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1-5534 


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0.3769 


37 


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1-2957 


1-5549 


1. 8140 


2.0732 


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38 


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1.2970 


1.5564 


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39 


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40 


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1.8192 


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2.3390 


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4i 


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42 


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45 


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1-3057 


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46 


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1.3069 


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47 


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1.3081 


1.5698 


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2.0930 


2-3547 


0.3812 


48 


c.2619 


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1 -0475 


1.3094 


I-57I2 


1. 8331 


2.0950 


2.3569 


0.3816 


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0.2621 


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0. 7864 


1.0485 


1. 3106 


L5727 


1.8348 


2.0970 


2.3591 


0.3820 


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0.2624 


0.5247 


0.7871 


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1.3118 


1.5742 


1.8366 


2.C990 


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0.2626 


0.5252 


0.7879 


1.0505 


1-3131 


1-5757 


1.8383 


2.1010 


2.3636 


0.3827 


52 


0.2629 


0.5257 


0.7886 


1.0514 


i-3 I 43 


1-577 2 


1.8400 


2.1029 


2.3658 


0.3831 


53 


0.2631 


0.5262 


0.7893 


1.0524 


I.3I55 


1.5787 


1.8418 


2.1049 


2.3680 


0.3835 


54 


0.2634 


0.5267 


0.7901 


1.0534 


1.3168 


1.5802 


1.8435 


2.1069 


2.3702 


0.3839 


55 


0.2636 


0.5272 


0.7908 


1.0544 


1.3180 


1.5816 


1.8452 


2.ic88 


2.3724 


0.3843 


56 


0.2638 


0.5277 


0.79I5 


1.0554 


1.3192 


1-5831 


1.8469 


2. 1 108 


2.3746 


0.3847 


57 


0.2641 


0.5282 


0.7923 


1.0564 


1.3205 


1.5846 


1.8487 


2.1128 


2.3769 


0.3851 


58 


0.2643 


0.5287 


0.7930 


I-0574 


1.3217 


1.5860 


1.8504 


2. 1 147 


2.3791 


0.3855 


I 9 


0.2646 


0.5292 


0.7938 


1.0584 


1.3230 


1.5875 


1.8521 


2. 1 167 


2.3813 


0.3859 


|6o 



120 DISTANCES. 


16° 


oo 


1 


3 


3 


4 


5 


6 


7 


8 


9 


a 


0.9227 


1.8455 


2.7682 


3.6909 


4.6i37 


5-5364 


6.4591 


7.3818 


8.3046 


1-3458 


OI 


0.9226 


1.8452 


2.7677 


3.6903 


4.6129 


5-5355 


6.4580 


7.3806 


8.3032 


1 1-3457 


02 


0.9224 


1.8448 


2.7673 


3.6897 


4.6121 


5-5345 


6.4569 


7-3794 


8.3018 


1-3456 


03 


0.9223 


1-8445 


2.7668 


3.6891 


1 4-6113 


5-5336 


6-4559 


7.378i 


8.3004 


1-3455 


04 


0.9221 


1.8442 


2.7663 


3.6884 


4.6006 


5-5327 


6.4548 


7-3769 


8.2990 


1-3454 


05 


0.9220 


1.8439 


2.7659 


3.6878 


4.6098 


5-53I7 


6-4537 


7-3757 


8.2976 


1-3453 


o5 


0.9218 


1.8436 


2.7654 


3.6872 


4.6090 


5.53o8 


6.4526 


7-3744 ! 8.2962 


1-345* 


o? 


0.9216 


I.8433 


2.7649 


3.6866 


4.60S2 


5.5299 


6.4515 


7-3732 


8.2948 


I-3450 


08 


0.9215 


1.8430 


2.7645 


3.6860 


4.6075 


5.5290 


6.4505 


7.37I9 


8.2934 


1-3449 


09 


0.9213 


1.8427 


2.7640 


3.6854 


4.6067 


5.5280 


6-4494 


7-3707 


8.2921 


1-3448 


10 


0.9212 


1.8424 


2.7636 


3.6847 


4.6059 


5.5271 


6.4483 


7.3695 


8.2907 


1.3447 


11 


0.9210 


1. 8421 


2.7631 


3.6841 


4.6051 


5-5262 


6.4472 


7.3682 8.2893 


1.3446 


12 


0.9209 


1.8417 


2.7626 


3-6835 


4.6044 


5-5252 


6.4461 


7.3670 ' 8.2878 


1-3445 


*3 


0.9207 


1. 8414 


2.7621 


3.6829 


4.6036 


5o243 


6.4450 


7.3657 ! 8.2864 


1 1-3444 


*4 


0.9206 


1. 841 1 


2.7617 


3.6822 


4.6028 


5-5234 


6-4439 


7.3645 8.2850 


■ 1-3442 


*5 


0.9204 


1.8408 


2.7612 


3.6816 


4.6020 


5-5224 


6.4428 


7.3632 | 8.2836 


'< 1.344* 


16 


0.9202 


1.8405 


2.7607 


3.6810 


4.6012 


5-5215 


6.4417 


7.3620 8.2822 


' I-3440 


17 


0.9201 


1.8402 


2.7603 


3.6804 


4.6005 


5-5205 


6.4406 


7.3607 1 8.2808 


1-3439 


18 


0.9199 


1.8399 


2.7598 


3.6797 


4-5997 


5-5196 


6-4395 


7-3595 8.2794 


1 1.3438 


*9 


0.9198 


1.8396 


2-7593 


3.679I 


4.5989 


5.5187 


6.4385 


7.3582 


8.2780 


i 1-3437 


20 


0.9196 


1.8392 


2.7589 


3-6785 


40981 


5.5I77 


6-4374 


7-3570 


8.2766 


1.3436 


21 


0.9195 


1.8389 


2.7584 


3.6779 


4-5973 


5.5168 


6.4363 


7.3557 


8.2752 


1-3435 


22 


0.9193 


1.8386 


2-7579 


3-6772 


40965 


5.5158 


6.4352 


7-3545 


8.2738 


1-3433 


2 3 


0.9192 


1.8383 


2-7575 


3.6766 j 


4.5958 


5.5I49 


6.4341 


7-3532 


8.2724 


1.3432 


24 


0.9190 


1.8380 


2.7570 


3.6760 


4-5950 


5.5140 


6.4330 


7-3519 


8.2709 


I.343I 


25 


0.9188 


1.8377 


2.7565 


3.6753 1 


4.5942 


5.5130 


6.4319 


7-3507 


8.2695 


I-3430 


26 


0.9187 


1.8374 


2.7560 


3-6747 | 


4-5934 


S-SI2I 


6.4307 


7-3494 


8.26S1 


1.3429 


27 


0.9185 


1.8370 


2.7556 


3-674I 


4.5926 


5.5m 


6.4296 


7.3482 


8.2667 


1.3428 


28 


0.9184 


1.8367 


2.7551 


3-6735 


4.5918 


5-5102' 


6.4285 


7-3469 


S.2653 


1.3427 


29 


0.9182 


1.8364 


2.7546 


3.6728 ; 


4-59io 


5-5092 


6.4274 


7.3456 


8.2639 


1.3425 


30 


0.9180 


1.8361 


2.7541 


3.6722 


4.5902 


5.5083 


6.4263 


7-3444 


8.2624 


1.3424 


3i 


0.9179 


1.8358 


2-7537 


3.6716 


4.5894 


5-5073 


6.4252 


7-3431 


8. 2610 


1.3423 


32 


0.9177 


1.8355 


2.7532 


3.6709 


4.5887 


5-5064 


6.4241 


7.3418 8.2596 


1.3422 


33 


c.9176 


1.8351 


2.7527 


3-6703 : 


4o879 


5-5054 


6.4230 


7.3406 8.25S1 1 


1.3420 


34 


0.9174 


1.8348 


2.7522 


3.6696 


4-5871 


5-5045 


6.4219 


7-3393 8.2567 j 


1.34*9 


35 


0.9*73 


1.834S 


2.7518 


3.6690 


4.5S63 


5-5035 


6.4208 


7.3380 1 S.2553 


1-34*8 


36! 


0.9171 


1.8342 


2.7513 


3.6684 


4o855 


5-5026 


6.4197 


7.3368 8.2539 


1-34*7 


37 


0.9169 


1.8339 


2.750S 


3-6677 


4-5S47 


5-5oi6 


6.41S6 


7-3355 S.2524 


1-34*6 


3S 


0.916S 


1.8336 


2.7503 


3.6671 


4o839 


5-5007 


6.4174 


7.3342 


8.2510 


1. 34*5 


39 


0.9166 


1.8332 


2.7499 


3.6665 


4-583I 


5-4997 


6.4163 


7.3329 


8.2496 


1.34*3 


40 


0.9165 


1.8329 


2-7494 


3.665s 


4-5823 


5.4988 


6.4152 


7.33I7 


8.2481 ! 


1.3412 


4i 


0.9163 


1.8326 


2.7489 


3.6652 


4.5815 


5-497S 


6.4141 


7.3304 


S.2467 j 


I-34** 


42 


0.9161 


1.8323 


2.7484 


3.6646 


40807 


5.496S 


6.4130 


7.3291 i S.2452 


I-34IO 


43 


0.9160 


1.8320 


2-7479 


3-6639 


4-5799 


5-4959 


6.411S 


7.3278 S.2438 


1.3409 


44 


0.9158 


1. 8316 


2-7475 


3-6633 


4.5791 


5-4949 


6.4107 


7-3265 


S.2424 


I.3407 


45 


o-9*57 


1-8313 


2.7470 


3.6626 


4-5783 


5-4939 


6.4096 


7.3253 


S.2409 


1.3406 


46 


0.9155 


1. 8310 


2.7465 


3.6620 ! 


4-5775 


5-4930 


6.40S5 


7-3240 


8.2395 


I-3405 


47 


0.9153 


1.S307 


2.7460 


3-6613 


4.5767 


5.4920 


0.4074 


7-3227 


8.2380 


I-3404 


48 


0.9152 


1.8304 


2-7455 


3.6607 


4-5759 


5.491* 


6.4062 


7-3214 


8.2366 


I-3403 


49 


0.9150 


1.S3CO 


2.7450 


3.6601 


4-5751 


5.4901 


6.4051 


7.3201 


8.2351 ; 


1.3402 


50 


0.9149 


1.S297 


2.7446 


3-6594 


4-5743 


5.4891 


6.4040 


7-3ISS 


S.2337 


1.3400 


5i 


0.9147 


1.8294 


2.7441 


3.65S8 


4.5735 


5.4SS2 


6.4029 


7-3176 


S.2322 


1-3399 


52 


0.9145 


1. 8291 


2.7436 


3-6 5 Si 


4.5727 


5.4S72 


6.4017 


7.3163 j 8.230S 1 


1.3398 


53 


0.9144 


1.8287 


2-743 1 


3-6575 


4.5719 


5.4S62 


6.4006 


7-3*50 


8.2293 


1-3397 


54 


0.9142 


1.S284 


2.7426 


3.6568 


4-57IO 


5-4S53 


6-3995 


7.3I37 


S.2279 


1-3395 


55 


0.9140 


1.S2S1 


2.7421 


3.6562 


4.5702 


5-4S43 


6.3983 


7-3 I2 4 


S.2264 


1-3394 


56; 


0.9139 


1.8278 


2.7417 


3-6555 


4.56Q4 


5-4S33 


6.3972 


7.3m 


S.2250 


1-3393 


57! 


0.9137 


1.S274 


2.7412 


3-6549 


4.56S6 


5-4S23 


6.3961 


7.3098 


^ 2235 


1.3392 


5* 


0.9136 


1.8271 


2.7407 


3-6542 ; 


4067S 


5-4Si4 


6-3949 


7.3085 


S.222I 


I-3390 


59 


0.9134 


1.S26S 


2. 7402 


3-6536 


4.5670 


5.4S04 


6.3938 


- 3c 72 S.2206 


1.3389 


60 


j 0.9132 


1.S265 


2-7397 


3.6530 


4.5662 


5.4794 


6.392 7. 


7.3059 8.2 192 


1.33JB8 



16° 


HEIGHTS. 121 


1 


3 


3 


4 


5 


6 


7 


8 


9 


b 

0.3859 


/ 

00 


0.2646 


0.5292 


0.7938 


1.0584 


1.3230 


I-5875 


1.8521 


2. 1 167 


2.3813 


0.2648 


0.5297 


o.7945 


I.0594 


T.3242 


1.5890 


1-8539 


2.1187 


2.3836 


0.3863 


01 


0.2651 


0.5302 


0.7952 


1.0603 


1.3254 


I-5905 


1.8556 


2.1206 


2.3857 


0.3867 


02 


0.2653 


0.5307 


0.7960 


1.0613 


1.3266 


1.5920 


1.8573 


2.1226 


2.3880 


0.3871 


03 


0.2656 


0.5311 


0.7967 


1.0623 


L3279 


1-5934 


1.8590 


2.1246 


2.3902 


0.3875 


04 


0.2658 


o.53 l6 


o.7975 


1-0633 


1. 3291 


1-5949 


1.8607 


2.1266 


2.3924 


0.3878 


05 


0.2661 


0.5321 


0. 7982 


1.0642 


I.3303 


1.5964 


1.8624 


2.1285 


2.3946 


0.3882 


06 


0.2663 


0.5326 


0.7989 


1.0652 


I.33i6 


1-5979 


1.8642 


2.1305 


2.3968 


0.3886 


07 


0.2666 


o.533i 


o.7997 


1.0662 


L3328 


1-5994 


1.8659 


2.1325 


2.3990 


0.3890 


08 


0.2668 


0.5336 


0.8004 


1.0672 


I-3340 


1.6008 


1.8676 


2.1344 


2.4012 


0.3894 


09 


0.2670 


o.534i 


0.801 1 


1.0682 


1-3352 


1.6023 


1.8693 


2.1364 


2.4034 


0.3898 


10 


0.2673 


0.5346 


0.8019 


1.0692 


I-3365 


1.6037 


1. 8710 


2.1383 


2.4056 


0.3902 


11 


0.2675 


o.535i 


0.8026 


1.0702 


1-3377 


1.6052 


1.8728 


2.1403 


2.4078 


0.3906 


12 


0.2678 


o.5356 


0.8033 


1.0711 


1.3389 


1.6067 


1-8745 


2.1422 


2.4100 


0.3910 


13 


0.2680 


0.5361 


0.8041 


1. 0721 


1.3401 


1.6082 


1.8762 


2.1442 


2.4123 


0.3914 


14 


0.2683 


0.5365 


0.8048 


1.0731 


1. 3414 


1.6096 


1.8779 


2.1462 


2.4145 


o-39 J 7 


15 


0.2685 


0.5370 


0.8056 


1.0741 


1.3426 


1.6111 


1.8796 


2.1482 


2.4167 


0.3921 


16 


0.2688 


o.5375 


0.8063 


1.0750 


L3438 


1. 6126 


1.8813 


2.1501 


2.4189 


0.3925 


17 


0.2690 


0.53S0 


0.8070 


1.0760 


i.345o 


1.6141 


1. 8831 


2.1521 


2.4211 


0.3929 


18 


0.2693 


0.5385 


0.8078 


1.0770 


I-3463 


i-6i55 


1.8848 


2.1540 


2.4233 


o.3933 


19 


0.2695 


0.5390 


0.8085 


1.0780 


1-3475 


1. 6170 


1.8865 


2.1560 


2.4255 


o.3937 


20 


0.2697 


o.5395 


0.8092 


1.0790 


I-3487 


1. 6184 


1.8882 


2.1579 


2.4277 


0.3941 


21 


0.2700 


0.5400 


0.8100 


1.0800 


1-3499 


1. 6199 


1.8899 


2.1599 


2.4299 


o.3945 


22 . 


0.2702 


0.5405 


0.8107 


1.0809 


i-35i2 


1. 6214 


1. 8916 


2.1618 


2.4321 


0.3949 


23 


0.2705 


0.5409 


0.8114 


1. 0819 


I-3524 


1.6228 


I-8933 


2.1638 


2-4343 


o.3953 


24 


0.2707 


0.5414 


0.8122 


1.0829 


1-3536 


1.6243 


1.8950 


2.1658 


2.4365 


0-3957 


25 


0.2710 


0.5419 


0.8129 


1.0838 


1-3548 


1.6258 


1.8967 


2.1677 


2.4387 


0.3960 


26 


0.2712 


0.5424 


0.8136 


1.0848 


i-356o 


1.6273 


1.8985 


2.1697 


2.4409 


0.3964 


27 


0.2715 


0.5429 


0.8144 


1.0858 


1-3573 


1.6287 


1.9002 


2.1710 


2.4431 


0.3968 


28 


0.2717 


o.5434 


0.8151 


1.0868 


1.3585 


1.6302 


1. 9019 


2.1736 


2-4453 


0.3972 


129 


0.2719 


o.5439 


0.8158 


1.0878 


!-3597 


1.6310 


1.9036 


2.1755 


2.4475 


0.3976 


30 


0.2722 


o.5444 


0.8165 


1.0887 


1.3609 


1-633* 


I.9C53 


2.1775 


2.4496 


0.3980 


3 1 


0.2724 


0.5448 


0.8173 


1.0897 


1. 3621 


I-6345 


1.9070 


2.1794 


2.4518 


0.3984 


32 


0.2727 


o.5453 


0.8180 


1.0907 


1-3634 


1.6361 


1.9087 


2.1814 


2.4540 


0.3988 


|33 


0.2729 


o.5458 


0.8187 


1. 0916 


1.3646 


1-6375 


1. 9104 


2.1833 


2.4562 


0.3992 


34 


0.2732 


0.5463 


0.8195 


1.0926 


1.3658 


1.6390 


1.9121 


2.1853 


2.4584 


0.3996 


35 


C.2734 


0.5468 


0.8202 


1.0936 


1.3670 


1.6404 


1.9138 


2.1872 


2.4606 


0.3999 


36 


0.2736 


o.5473 


0.8209 


1.0946 


1.3682 


1.6418 


I-9I55 


2.1891 


2.4628 


0.4003 


37 


0.2739 


o.5478 


0.8216 


I.0955 


1.3694 


I-6433 


1.9172 


2.1911 


2.4650 


0.4007 


38 


0.2741 


0.5483 


0.8224 


1.0965 


1.3706 


1.6448 


1. 9189 


2. 1930 


2.4672 


0.401 1 


39 


0.2744 


0.5487 


0.8231 


I.0975 


I-37I9 


1.6462 


1.9206 


2.1950 


2.4693 


0.4015 


40 


0.2746 


0.5492 


0.8238 


1.0984 


1-373* 


1.6477 


1.9223 


2. 1969 


2.4715 


0.4019 


4i 


0.2749 


o.5497 


0.8246 


1.0994 


1-3743 


1. 6491 


1.9240 


2.1988 


2.4737 


c.4023 


42 


0.2751 


0.5502 


0.8253 


1. 1004 


1-3755 


1.6506 


1-9257 


2.2008 


2-4759 


c.4027 


43 


0.2753 


0.5507 


0.8260 


1.1014 


1.3767 


1.6520 


1.9274 


2.2027 


2.4781 


0.4031 


44 


0.2756 


0.5512 


0.8267 


1. 1023 


1-3779 


1-6535 


1. 9291 


2.2046 


2.4802 


0.4035 


45 


0.2758 


0.5516 


0.8275 


1. 1033 


i-379i 


1.6549 


1.9308 


2.2066 


2.4824 


0.4039 


46 


0.2761 


0.5521 


0.8282 


1. 1043 


1-3803 


1.6564 


1.9325 


2. 2085 


2.4846 


0.4042 


47 


0.2763 


0.5526 


0.8289 


1. 1052 


1.3815 


1.6579 


1-9342 


2.2105 


2.4868 


0.4046 


48 


0.2766 


o.553i 


0.8297 


1. 1062 


1.3828 


1.6593 


J-9359 


2.2124 


2.4890 


0.4050 


49 


0.2768 


C5536 


0.8304 


1. 1072 


1.3840 


1.6607 


1-9375 


2.2143 


2.491 1 


0.4054 


50 


0.2770 


o.554i 


0.831 1 


1.1081 


1.3852 


1.6622 


1.9392 


2.2163 


2-4933 


0.4058 


5 1 


0.2773 


0.5546 


0.8318 


1.1091 


1.3864 


1.6637 


1.9409 


2.2182 


2-4955 


0.4062 


S 2 


0.2775 


0.5550 


0.8326 


I.IIOI 


1.3876 


1.6651 


1.9426 


2.2202 


2-4977 


0.4066 


53 


0.2778 


o.5555 


0.8333 


1. mo 


1.3888 


1.6666 


1-9443 


2.2221 


2.4998 


0.4070 


54 


0.2780 


0.5560 


0.8340 


1.1120 


1.3900 


1.6680 


1.9460 


2.2240 


2.5020 


0.4074 


55 


0.2782 


0.5565 


0.8347 


1.113° 


1.3912 


1.6695 


1-9477 


2.2259 


2.5042 


0.4078 


56 


0.2785 


o.557o 


0.8354 


1. 1 139 


1.3924 


1.6709 


1.9494 


2.2278 


2.5063 


0.4081 


57 


0.2787 


o.5574 


0.8362 


1. 1149 


1.3936 


1.6723 


1.9510 


2.2298 


2.5085 


0.4085 


58 


0.2790 


o.5579 


0.8369 


1.1158 


1.3948 


1.6738 


1-9527 


2.2317 


2.5107 


0.4089 


59 


0*2792 


0.5584 


0.8376 


1. 1 168 


1.3960 


1.6752 


1-9544 


2.2337 


2.5129 


0.4093 


60 



122 DISTANCES. 17° 


oo 


1 


8 


3 


4 


5 


6 


7 


8 


9 


a 


0.9132 


1.8265 


2.7397 


3.6530 


4.5662 


5-4794 


6.3927 


7.3059 


8.2192 


1.338S 


OI 


0.9131 


1.8262 


2.7392 


3-6523 


4-5654 


5.4785 


6-39*5 


7.3046 


8.2177 


1-3387 


02 


0.9129 


1.8258 


2.7387 


3-6517 


4.5646 


5-4775 


6.3904 


7.3033 


8.2162 


1-3385 


03 


0.9127 


1.8255 


2.7382 


3-6510 


4.5637 


5.4765 


6.3892 


7.3020 


8.2147 


1-3384 


04 


0.9126 


1.8252 


2.7378 


3-6503 


4.5629 


5-4755 


6.3881 


7.3007 


8.2133 


1.3383 


05 


0.9124 


1.8248 


2-7373 


3.6497 


4.5621 


5-4745 


6.3870 


7.2994 


8.2118 


1.3382 


06 


0.9123 


1.8245 


2.7368 


3.6490 


4.56i3 


5-4736 


6.3858 


7.2981 


8.2103 


1. 338i 


07 


0.9121 


1.8242 


2.7363 


3.6484 


4-5605 


5.4726 


6.3847 


7.2968 


8.2089 


1-3379 


08 


0.9119 


1.8239 


2.7358 


3-6477 


4-5597 


5-47i6 


6.3835 


7-2955 


8.2074 


1.3378 


09 


0.9118 


1-8235 


2-7353 


3<647i 


4-5589 


5.4706 


6.3824 


7.2942 


8.2059 


1-3377 


10 


0.9116 


1.8232 


2.7348 


3.6464 


4-558o 


5.4696 


6.3812 


7.2929 


8.2045 


1.33/6 


II 


0.9114 


1.8229 


2.7343 


3.6458 


4-5572 


5.4687 


6.3801 


7.2915 


8.2030 


1-3375 


12 


0.91 13 


1.8226 


2.7338 


3-6451 


4-5564 


5.4677 


6.3789 


7.2902 


8.2015 


1-3373 


13 


0.9111 


1.8222 


2-7333 


3-6445 


4o556 


5.4667 


6.3778 


7.2889 


8.2000 


1.3372 


14 


0.9109 


1. 8219 


2.7328 


3.6438 


4-5547 


5-4657 


6.3766 


7.2876 


8.1985 


I-337I 


15 


0.910S 


1. 8216 


2.7324 


3.643I 


4-5539 


5-4647 


6-3755 


7.2863 


8.1971 


I-3370 


16 


0.9106 


1.8212 


2.7319 


3-6425 


4-5531 


5-4637 


6-3743 


7.2850 


8.1956 


1.3368 


17 


0.9105 


1.8209 


2. 73*4 


3.6418 


4-5523 


5.4627 


6.3732 


7.2836 


8.1941 


1-3367 


18 


0.9103 


1.8206 


2.7309 


3.6412 


4-5515 


5-4617 


6.3720 


7.2823 


8.1926 


1.3366 


J 9 


0.9101 


1.8203 


2.7304 


3-6405 


4-55o6 


5.4608 


6.3709 


7.2810 


8.1911 


1-3365 


20 


0.9100 


1. 8199 


2.7299 


3-6398 


4.5498 


5.4598 


6.3697 


7.2797 


8.1897 


1-3364 


21 


0.9098 


1. 8196 


2.7294 


3.6392 


4.5490 


5.4588 


6.3686 


7.2784 


8.1882 


1.3362 


22 


0.9096 


1.8193 


2. 7289 


3.6385 


4.5482 


5.4578 


6.3674 


7.2770 


8.1867 


i.336i 


23 


0.9095 


1. 8189 


2.7284 


3.6379 


4-5473 


5.4568 


6.3662 


7-2757 


8.1852 


1.3360 


24 


0.9093 


1.8186 


2.7279 


3-6372 


4.5465 


5.4558 


6.3651 


7-2744 


8.1837 


1-3359 


25 


0.9091 


1.8183 


2.7274 


3-6365 


4-5457 


5.4548 


6.3639 


7.273I 


8.1822 


1.3358 


26 


0.9090 


1. 8179 


2.7269 


3.6359 


4.5448 


5.4538 


6.3628 


7.2717 


8.1807 


1-3357 


27 


0.9088 


1.8176 


2.7264 


3-6352 


4.5440 


5.4528 


6.3616 


7.2704 


8.1792 


1-3355 


28 


0.9086 


I-8I73 


2.7259 


3.6345 


4.543 2 


5.45i8 


6.3604 


7.2691 


8.1777 


1-3354 


29 


0.9085 


1. 8169 


2.7254 


3.6339 


4.5423 


5-4508 


6-3593 


7.2678 


8.1762 


1-3353 


30 


0.9083 


1.8166 


2.7249 


3.6332 


4.54I5 


5.4498 


6.35S1 


7.2664 


8.1747 


1-3352 


31 


0.9081 


1.8163 


2.7244 


3.6325 


4.5407 


5.4488 


6.3570 


7.2651 


S.1732 


I-3350 


32 


0.9080 


1.8159 


2.7239 


3-6319 


4.5398 


5-4478 


6.3558 


7.2637 


8.1717 


1-3349 


33 


0.9078 


1.8156 


2.7234 


3- 6 3 12 


4-5390 


5.4468 


6.3546 


7.2624 


8. 1702 


I-334S 


34 


0.9076 


I.8I53 


2.7229 


3-6305 


4.5382 


5.4458 


6-3534 


7.2611 


S.16S7 


1-3347 


35 


0.9075 


1. 8149 


2.7224 


3.6299 


4-5373 


5.4448 


6.3523 


7-2597 


8.1672 


1-3346 


36 


0.9073 


1. 8146 


2.7219 


3.6292 


4-5365 


5.4438 


6.351 1 


7.2584 


S.1657 


1-3344 


37 


0.9071 


1.8143 


2.7214 


3.6285 


4-5357 


5.4428 


6-3499 


7.257I 


8.1642 


1-3343 


38 


0.9070 


1.8139 


2.7209 


3.6279 


4.5348 


5.441S 


6.348S 


7-2557 


8.1627 


1-3342 


39 


0.9068 


1.8136 


2.7204 


3.6272 


4-5340 


5.4408 


6.3476 


7-2544 


8.1612 


I-334I 


40 


0.9066 


I.8I33 


2.7199 


3.6265 


4-5332 


5-4398 


6.3464 


7.2530 


8.1597 


1-3339 


4 1 


0.9065 


1. 8129 


2.7194 


3.6258 


4.5323 


5-438S 


6.3452 


7-25I7 


S'.i 5 8i 


1.3338 


42 


0.9063 


1. 8126 


2.7189 


3.6252 


4.53I5 


5-4378 


6.3441 


7-2503 


8.1566 


1-3337 


43 


0.9061 


1. 8122 


2.7184 


3-6245 


4.53o6 


5-4367 


6.3429 


7.2490 


S.1551 


1-3336 


44 


0.9060 


1.8119 


2.7179 


3.6238 


4.5298 


5-4357 


6.3417 


7.2476 


S.1536 


1-3335 


45 


0.9058 


1.8116 


2.7174 


3.6231 


4.5289 


5-4347 


6.3405 


7-2463 


S.1521 


1-3333 


46 


0.9056 


1.8112 


2.7169 


3.6225 


4.5281 


5-4337 


6-3393 


7-2449 


S.1506 


1.3332 


47 


0.9054 


1. 8109 


2.7163 


3.621S 


4-5272 


5-4327 


6.33S1 


7-2436 


8. 1490 


I-333I 


48 


0-9053 


1. 8106 


2.7158 


3.621 1 


4.5264 


5-4317 


6.3370 


7.2422 


8.1475 


I-3330 


49 


0.9051 


1. 8102 


2.7153 


3.6204 


4-5256 


5.4307 


6.335S 


7.2409 


8.1460 


L3329 


50 


0.9049 


1.8099 


2.7148 


3.619S 


4.5247 


5.4297 


6.3346 


7.2395 


8.1445 


1-3327 


5i 


0.9048 


1.8095 


2.7143 


3-6191 


4.5239 


5.42S6 


6-3334 


7-2382 


S.1430 


1-3326 


52 


0.9046 


1.8092 


2.7138 


3.6184 


4-5230 


5-4276 


6.3322 


7-236S 


S.1414 


I.3325 


53 


0.9044 


1.8089 


2.7133 


3.6i77 


4.5222 


5.4266 


6.3310 


7.2355 


S.1399 


I.3324 


54 


0.9043 


1.8085 


2.7128 


3-6171 


4-5213 


5-4256 


6.3298 


7.2341 


S.13S4 


I.3323 


55 


0.9041 


1.S082 


2.7123 


3.6164 


4.5205 


5.4246 


6.32S7 


7.2327 


S.136S 


I-332I 


56 


0.9039 


1.8078 


2.7118 


3-6i57 


4.5196 


5.4235 


6.3275 


7-2314 


S.I353 


1.3320 


57 


0.9038 


1.8075 


2.7113 


3-6150 


4.5188 


5-4225 


6.3263 


7.2300 


S.133S 


i.33i9 


58 


0.9036 


1.8072 


2.7107 


3-6i43 


4.5I79 


5-4215 


6.3251 


7.2287 


8.1322 


i-33iS 


59 


0.9034 


1.8068 


2.7102 


3-6i37 


4-5I7I 


5-4205 


6.3239 


7-2273 


S.1307 


i.33i6 


60 


0.9032 


1.8065 


2.7097 


3.6130 


4.5162 


5.4I95 


6.3227 


7.2259 


S.1292 


I.33I5 



17° 




HEIGHTS. 123 


l 


2 


3 


4 


5 


6 


7 


8 


9 


b 


/ 
00 


0.2792 


0.5584 


0.8376 


1.1168 


1.3960 


1.6752 


1-9544 


2.2337 


2.5129 


0.4093 


o.2794 


0.5589 


0.8383 


1.1178 


1.3972 


1.6767 


1.9561 


2.2356 


2.5150 


0.4097 


01 


o.2797 


o.5594 


0.8391 


1.1187 


1.3984 


1. 6781 


1.9578 


2.2375 


2.5172 


0.4101 


02 


0.2799 


o.5599 


0.8398 


1.1197 


1.3996 


1.6796 


1-9595 


2.2394 


2-5T93 


0.4105 


03 


0.2802 


0.5603 


0.8405 


1.1207 


1.4008 


1. 6810 


1. 9612 


2.2414 


2.5215 


0.4109 


04 


0.2804 


0.5608 


0.8412 


1.1216 


1 .4020 


1.6825 


1.9629 


2.2433 


2.5237 


0.4113 


c 5 


0.2806 


0.5613 


0.8419 


1. 1226 


1.4032 


1.6839 


1-9645 


2.2452 


2.5258 


0.4116 


06 


0.2809 


0.5618 


0.8427 


1. 1236 


1.4044 


1.6853 


1.9662 


2.2471 


2.5280 


0.4120 


07 


0.2811 


0.5623 


0.8434 


1. 1245 


1.4056 


1.6868 


1.9679 


2.2490 


2.5302 


0.4124 


08 


0.2814 


0.5627 


0.8441 


1.1255 


1.4068 


1.6882 


1.9696 


2.2510 


2.5323 


0.4128 


09 


0.2816 


0.5632 


0.8448 


1. 1264 


1.4080 


1.6897 


I.97I3 


2.2529 


2-5345 


0.4132 


10 


0.2818 


o.5 6 37 


O.8455 


1. 1274 


1.4092 


1.6911 


1.9729 


2.2548 


2.5366 


0.4136 


11 


0.2821 


0.5642 


0.8463 


1. 1284 


1.4104 


1.6925 


1.9746 


2.2567 


2.5388 


0.4140 


12 


0.2823 


0.5647 


0.8470 


1. 1293 


1.4116 


1.6940 


1.9763 


2.2586 


2.5409 


0.4144 


13 


0.2826 


0.5651 


0.8477 


1. 1303 


1.4128 


1.6954 


1.9780 


2.2606 


2.5431 


0.4148 


14 


0.2828 


0.5656 


0.8484 


1.1312 


1.4140 


1.6969 


1-9797 


2.2625 


2.5453 


0.4151 


15 


0.2830 


0.5661 


0.8491 


1. 1322 


1.4152 


1.6983 


1.9813 


2.2644 


2-5474 


0.4155 


16 


0.2833 


0.5666 


0.8499 


1. 1332 


1.4164 


1.6997 


1.9830 


2.2663 


2.5496 


0.4159 


17 


0.2835 


0.5670 


0.8506 


1.1341 


1.4176 


1. 701 1 


1.9847 


2.2682 


2.5517 


0.4163 


18 


0.2838 


0.5675 


0.8513 


1.1351 


1.4188 


1.7026 


1.9863 


2.2701 


2-5539 


0.4167 


19 


0.2840 


0.5680 


0.8520 


1. 1360 


1.4200 


1.7040 


1.9880 


2.2720 


2.5560 


0.4171 


20 


0.2842 


0.5685 


0.8527 


1. 1370 


1. 4212 


I-7054 


1.9097 


2.2739 


2.5582 


o.4i75 


21 


0.2845 


0.5690 


0.8534 


1 -1379 


1.4224 


1.7069 


I-99I4 


2.2758 


2.5603 


c.4179 


22 


0.2847 


0.5694 


0.8542 


1. 1389 


1.4236 


1.7083 


1.9930 


2.2778 


2.5625 


0.4183 


23 


0.2850 


0.5699 


0.8549 


1. 1398 


1.4248 


1.7098 


1.9947 


2.2797 


2.5646 


0.4186 


24 


0.2852 


0.5704 


0.8556 


1. 1408 


1.4260 


1.7112 


1.9964 


2.2816 


2.5668 


0.4190 


25 


0.2854 


0.5709 


0.8563 


1.1417 


1.4272 


1. 7126 


1.9980 


2.2834 


2.5689 


0.4194 


26 


0.2857 


0.57I3 


0.8570 


1. 1427 


1.4284 


1. 7140 


1.9997 


2.2854 


2.5710 


0.4198 


27 


0.2859 


0.5718 


0.8577 


1. 1436 


1.4296 


1. 7155 


2.0014 


2.2873 


2.5732 


0.4202 


28 


0.2861 


o.5723 


0.8584 


1. 1446 


I.4307 


1. 7169 


2.0030 


2.2892 


2-5753 


0.4208 


29 


0.2864 


0.5728 


0.8592 


1. 1456 


I-43I9 


1. 7183 


2.0047 


2.2911 


2-5775 


0.4210 


30 


0.2866 


0.5732 


0.8599 


1. 1465 


I.433I 


1. 7197 


2x063 


2.2930 


2.5796 


0.4214 


3 1 


0.2869 


o.5737 


0.8606 


1. 1474 


1-4343 


1. 7212 


2.0080 


2.2949 


2.5818 


0.4217 


32 


0.2871 


o.5742 


0.8613 


1. 1484 


L4355 


1.7226 


2.0097 


2.2968 


2.5839 


0.4221 


33 


0.2873 


o.5747 


0.8620 


1. 1494 


1.4367 


1.7240 


2.0114 


2.2987 


2.5861 


0.4225 


34 


0.2876 


0.5752 


0.8627 


1-1503 


1-4379 


1-7255 


2.0131 


2.3006 


2.5882 


0.4229 


3 I 


0.2878 


0.5756 


0.8634 


1.1512 


1.4390 


1.7269 


2.0147 


2.3025 


2.5903 


0.4233 


36 


0.2880 


0.5761 


0.8641 


1. 1522 


1.4402 


1.7283 


2.0163 


2.3044 


2.5924 


0.4237 


37 


0.2883 


0.5766 


0.8649 


i. 1532 


1.4414 


1.7297 


2.0180 


2.3063 


2.5946 


0.4241 


38 


0.2885 


0.5770 


0.8656 


1.1541 


1.4426 


t-73" 


2.0196 


2.3082 


2.5967 


0.4245 


39 


0.2888 


0.5775 


0.8663 


1. 1550 


1.4438 


1.7326 


2.0213 


2.3101 


2.5988 


0.4249 


40 


0.2890 


0.5780 


0.8670 


1.1560 


1.4450 


1 -734Q 


2.0230 


2.3120 


2.6010 


0.4252 


4i 


0.2892 


0.5785 


0.8677 


1. 1569 


1.4462 


1-7354 


2.0246 


2.3139 


2.6031 


0.4256 


42 


0.2895 


0.5789 


C.8684 


I - I 579 


1.4474 


1.7368 


2.0263 


2.3158 


2.6052 


0.4260 


43 


0.2897 


o.5794 


0.8691 


1. 1588 


1.4485 


1.7383 


2.0280 


2.3177 


2.6074 


0.4264 


44 


0.2899 


0.5799 


0.8698 


1. 1598 


1.4497 


1-7397 


2.0296 


2.3196 


2.6095 


0.4268 


45 


0. 2902 


0.5804 


0.8705 


1. 1607 


1.4509 


1. 741 1 


2.0313 


2.3215 


2.6116 


0.4272 


46 


0.2904 


0.5808 


0.8713 


1.1617 


1.4521 


1-7425 


2.0329 


2.3233 


2.6138 


0.4276 


47 


0.2907 


0.5813 


0.8720 


1. 1626 


1-4533 


1-7439 


2.0345 


2.3252 


2.6159 


0.4280 


48 


0.2909 


0.5818 


0.8727 


1. 1636 


1-4544 


1-7453 


2.0362 


2.3271 


2.6180 


0.4283 


49 


0.291 1 


0.5823 


0.8734 


1. 1645 


1.4556 


1.7468 


2.0379 


2.3290 


2.6202 


0.4287 


50 


0.2914 


0.5827 


0.8741 


1. 1654 


1.4568 


1.7482 


2.0395 


2.3309 


2.6223 


0.4291 


5i 


0.2916 


0.5832 


0.8748 


1. 1664 


1.4580 


1.7496 


2.0412 


2.3328 


2.6244 


0.4295 


52 


0.2918 


0.5837 


0.8755 


1. 1673 


1.4591 


1. 7510 


2.0428 


2.3346 


2.6265 


0.4299 


53 


0.2921 


0.5841 


0.8762 


1. 1683 


1.4603 


1.7524 


2.0445 


2.3365 


2.6286 


0.4303 


54 


0.2923 


0.5846 


0.8769 


1. 1692 


1.4615 


1-7538 


2.0461 


2.3384 


2.6307 


0.4307 


55 


0.2925 


0.5851 


0.8776 


1. 1702 


1.4627 


1-7552 


2.0478 


2.3403 


2.6329 


0.431 1 


56 


0.2928 


0.5856 


0.8783 


1.1711 


1.4639 


1-7567 


2.0495 


2.3422 


2.6350 


0.4315 


57 


0.2930 


0.5860 


0.8790 


1. 1720 


1.4651 


1. 7581 


2.0511 


2.3441 


2.6371 


0.4318 


58 


0.2932 


0.5865 


0.8797 


1. 1730 


1.4662 


1-7595 


2.0527 


2.3460 


2.6392 


0.4322 


59 


0.2935 


0.5870 


0.8804 


1. 1739 


1.4674 


1.7609 


2.0544 


2.3478 


2.6413 


0.4326 


60 



124 


DISTANCES. 18° 


oo 


1 


3 


3 


4 


5 


6 


7 


8 


9 


a 


0.9032 


1.8065 


2.7097 


3.6130 


4.5162 


5-4*95 


6.3227 


7.2259 


8. 1292 


I-33I5 


OI 


0.9031 


1. 8061 


2.7092 


3-6123 


4-5*54 


5.4184 


6.3215 


7.2246 


8.1276 


I-33I4 


02 


0.9029 


1.8058 


2.7087 


3.6116 


4.5I45 


5.4174 


6.3203 


7.2232 


8.1261 


1.3312 


03 


0.9027 


1.8055 


2.7082 


3.6109 


4-5136 


5.4164 


6.3191 


7.2218 


8.1246 


i-33" 


04 


0.9026 


1. 8051 


2.7077 


3.6102 


4.5128 


5.4153 


6.3179 


7.2205 


8. 1230 


1.3310 


05 


0.9024 


1.8048 


2.7072 


3-6095 


4.5119 


5.4I43 


6.3167 


7.2191 


8.1215 


I-3309 


06 


0.9022 


1.8044 


2.7066 


3.6089 


4.5111 


5-4I33 


6.3155 


7.2177 


8. 1 199 


1.3308 


07 


0.9020 


1. 8041 


2.7061 


3.6082 


4.5102 


5-4 I2 3 


6.3143 


7.2163 


8.1184 


1.3306 


08 


0.9019 


1.8037 


2.7056 


3-6o75 


4.5094 


5.4112 


6.3131 


7.2150 


8. 1 168 


1.3305 


09 


0.9017 


1.8034 


2.7051 


3.6068 


4-5085 


5.4102 


6.3119 


7.2136 


8. 1 153 


1-3304 


10 


0.9015 


1. 8031 


2. 7046 


3-6o 4 6i 


4.5076 


5.4092 


6.3107 


7.2122 


8. 1 138 


1.3302 


II 


0.9014 


1.8027 


2. 7041 


3-6054 


4.5068 


5.4081 


6.3095 


7.2108 


8.1122 


1.3301 


12 


0.9012 


1.8024 


2.7035 


3-6047 


4-5059 


5.4071 


6.3083 


7.2095 


8.1106 


1-3300 


J 3 


0.9010 


1.8020 


2.7030 


3.6040 


4-5050 


5.4061 


6.3071 


7.2081 


8.1091 


1.3298 


14 


0.9008 


1. 8017 


2.7025 


3-6033 


4.5042 


5.4050 


6.3059 


7.2067 


8.1075 


I-3297 


15 


0.9007 


1. 8013 


2. 7020 


3.6027 


4.5033 


5.4040 


6.3046 


7-2053 


8.1060 


1.3296 


16 


0.9005 


1. 8010 


2.7015 


3.6020 


4-5025 


5.4029 


6.3034 


7.2039 


8.1044 


1.3294 


*7 


0.9003 


1.8006 


2.7010 


3-6013 


4.5016 


5-4019 


6.3022 


7.2025 


8.1029 


1-3293 


18 


0.9001 


1.8003 


2.7004 


3.6006 


4.5007 


5.4009 


6.3010 


7.2012 


8.1013 


1.3292 


x 9 


0.9000 


1.7999 


2.6999 


3-5999 


4-4999 


5.3998 


6.2998 


7.1998 


8.0998 


1.3291 


20 


0.8998 


1.7996 


2.6994 


3-5992 


4-499° 


5-3988 


6.2986 


7.1984 


8.0982 


1.3289 


21 


0.8996 


1-7993 


2.6989 


3-5985 


4.4981 


5-3978 


6.2974 


7.1970 


8.0966 


1.3288 


22 


0.8995 


1.7989 


2.6984 


3-5978 


4-4973 


5-3967 


6.2962 


7-I956 


8.0951 


1.3287 


23 


0.8993 


1.7986 


2.6978 


3-5971 


4.4964 


5-3957 


6.2949 


7.1942 


8.0935 


1.3285 


24 


0.8991 


1.7982 


2.6973 


3-5964 


4-4955 


5.3946 


6.2937 


7.1928 


8.0919 


1.3284 


25 


0.8989 


1.7979 


2.6968 


3-5957 


4.4946 


5-3936 


6.2925 


7-I9I4 


8.0904 


1.3283 


26 


0.8988 


1-7975 


2.6963 


3- 595o 


4.4938 


5-3925 


6.2913 


7.1900 


8.0888 


1.3281 


27 


0.8986 


1.7972 


2.6957 


3-5943 


4.4929 


5-3915 


6.2901 


7.1886 


8.0872 


1.3280 


28 


0.8984 


1.7968 


2.6952 


3.5936 


4.4920 


5-3904 


6.2888 


7-1873 


8.0857 


1.3279 


29 


0.8982 


1-7965 


2.6947 


3-5929 


4.4912 


5-3894 


6.2876 


7-1859- 


8.0841 


1.3278 


30 


0.8981 


1. 7961 


2.6942 


3.5922 


4.4903 


5.3884 


6.2864 


7-^845 


8.0825 


1.3276 


3i 


0.8979 


1.7958 


2.6937 


3-59 I 5 


4.4894 


5-3873 


6.2852 


7-1831 


8.0810 


1.3275 


32 


0.8977 


1-7954 


2.6931 


3.59o8 


4.4885 


5.3862 


6.2840 


7-1817 


8.0794 


1.3274 


33 


0.8975 


i- 795i 


2.6926 


3-590I 


4.4877 


5-3852 


6.2S27 


7.1S03 


8.0778 


1.3272 


34 


0.8974 


1-7947 


2.6921 


3-5894 


4.4868 


5-3841 


6.2S15 


7.1789 


8. 0762 


1-3271 


35 


0.8972 


1-7944 


2.6915 


3.5887 


4-4859 


5.3S31 


6.2S03 


7-1775 


8.0746 


1.3269 


36 


0.8970 


1.7940 


2.6910 


3.5880 


4.4850 


5.3820 


6.2790 


7.1760 


8.0731 


1.3268 


37 


0.8968 


1-7937 


2.6905 


3-5873 


4.4842 


5-3Sio 


6.277S 


7.1746 


8.0715 


1.3267 


38 


0.8967 


1-7933 


2.6900 


3-5866 


4-4S33 


5-3799 


6.2766 


7.T732 


8.0699 1 


1.3265 


39, 


0.8965 


I-7930 


2.6894 


3-5859 


4.4824 


5-3789 


6.-754 


7.171S 


S.0683 ! 


1.3264 


40 


0.8963 


1.7926 


2.6S89 


3-5852 


4.4815 


5.3778 


6.2741 


7.1704 


8.0667 ! 


1.3263 


41 


0.8961 


1.7923 


2.6S84 


3-5845 


4.4S06 


5-3768 


6.2729 


7.1690 


8.0651 


1.3262 


42 


0.8960 


I-79I9 


2.6S79 


3-5S3S 


4.4798 


5-3757 


6.2717 


7.1676 


S.0636 


1.3260 


43 


0.8958 


I-79I5 


2.6873 


3-583t 


4.4789 


5-3746 


6.2704 


7.1662 


8. 0620 


1.3259 


44| 


0.8956 


1. 7912 


2.6868 


3-5S24 


4.47S0 


5-3736 


6.2692 


7. 164S 


S.0604 


1-3258 


451 


o-S954 


1.7908 


2.6S63 


3.5S17 


4-4771 


5-37-5 


6.2679 


7-1634 


S.05SS 


L3257 


46 


0.8952 


I-79C5 


2.6S57 


3.5810 


4.4762 


5-3715 


6.2667 


7.1619 


8.0572 ! 


1.3255 


47 


0.S951 


1. 7901 


2.6S52 


3-5803 


4-4753 


5-3704 


6.2655 


7.1605 


8.0556 


1-3254 


48 


0.8949 


1.7898 


2.6S47 


3-5796 


4-4744 


5-3693 


6.2642 


7-i59i 


8. 0540 


1-3253 


49 


0.8947 


1.7894 


2.6841 


3-5789 


4.4736 


5.36S3 


6.2630 


7-1577 


S.0524 


1-3251 


5° 


0.8945 


1. 7891 


2.6S36 


3.578i 


4.4727 


5-3672 


6.2618 


7-1563 


S.050S 


1-3250 


5i 


0.8944 


1.7SS7 


2.6831 


3-5774 


4.4718 


5.3661 


6.2605 


7.1549 


S.0492 


1-3249 


52 


0.8942 


1.7SS4 


2.6825 


3-5767 


4.47C9 


5-3651 


6.2593 


7-1534 


S.0476 


L3247 


53 


0.8940 


1.7880 


2.6S20 


3-5760 


4.4700 


5.3640 


6.2580 


7.1520 


8.0460 ; 


1.3246 


54 


0.8938 


1.7876 


2.6815 


3-5753 


4.4691 


5.3629 


6.256S 


7.1506 


S.0444 


1-3245 


55 


0.8936 


1.7873 


2.6809 


3.5746 


4.46S2 


5.3619 


6.2555 


7.1492 


S.C42S 


1-3243 


56 


0.8935 


1.7S69 


2.6S04 


3-5739 


4.4673 


5.360S 


6.2543 


7-1477 


S.0412 


1-3242 


57 


0.S933 


1.7866 


2.6799 


3.5732 


4.4664 


5-3597 


6.2530 


7-1463 


S.0396 J 


1. 3241 


58 


0.8931 


1.7S62 


2.6793 


3.5724 


4.4656 


5.35S7 


6.251S 


7.1449 


8.0380 


1.3 239 


59 


0.8929 


L7S59 


2.67SS 


3.57I7 


4-4647 


5.3576 


6.2505 


7-1435 


S.0364 


1.3338 


60 


0.892S 


1.7855 


2.6783 


3.57io 


4.463S 


5.3565 


6.2493 


7- T 4 2 o 


S.0348 | 1.3237 



18° 




HEIGHTS. 125 


1 


2 


3 


4 

i- 1739 


5 


6 


7 


8 


9 


b 


00 


Q.2935 


0.5870 


0.8804 


1.4674 


1.7609 


2.0544 


2.3478 


2.6413 


0.4326 


0.2937 


0.5874 


0.881 1 


1. 1749 


1.4686 


1.7623 


2.0560 


2-3497 


2.6434 


0.4330 


01 


0.2940 


0.5879 


0.8819 


1. 1758 


1.4698 


1.7637 


2.0577 


2.3516 


2.6456 


o.4334 


02 


0.2942 


o. 5 88 4 


0.8826 


1. 1768 


1.4709 


1. 7651 


2.0593 


2-3535 


2.6477 


0.4338 


03 


o.2944 


0.5888 


0.8833 


1.1777 


1.4721 


1.7665 


2.0609 


2-3554 


2.6498 


0.4342 


,°4 


0.2947 


o.5893 


0.8840 


1. 1786 


1-4733 


1.7679 


2.0626 


2-3573 


2.6519 


0.4346 


l°§ 


o.2949 


o. 5 8 9 8 


0.8847 


1. 1796 


1.4744 


1.7693 


2.0642 


2.3591 


2.6540 


o.4349 


06 


0.2951 


0.5902 


0.8854 


1. 1805 


1-4756 


1.7707 


2.0659 


2.3610 


2.6561 


0-4353 


,07 


o.2954 


0.5907 


0.8861 


1.1814 


1.4768 


1.7721 


2.0675 


2.3629 


2.6582 


o.4357 


I08 


0.2956 


0.5912 


0.8868 


1. 1824 


1.4780 


1-7735 


2.0691 


2.3647 


2.6603 


0.4361 


09 


0.2958 


o.59i7 


0.8875 


1. 1833 


1.4791 


I-7750 


2.0708 


2.3666 


2.6625 


0.4365 


10 


0.2961 


0.5921 


0.8882 


1. 1842 


1.4803 


1.7764 


2.0724 


2.3685 


2.6645 


0.4369 


11 


0.2963 


0.5926 


0.8889 


1. 1852 


i-48i5 


1.7778 


2.0740 


2.3703 


2.6666 


o.4373 


12 


0.2965 


o.593i 


0.8896 


1.1861 


T.4826 


1.7792 


2.0757 


2.3722 


2.6688 


o.4377 


13 


0.2968 


o.5935 


0.8903 


1. 1870 


1.4838 


1.7806 


2.0773 


2.3741 


2.6709 


0.4380 


14 


0.2970 


0.5940 


0.8910 


1. 1880 


1.4850 


1.7820 


2.0790 


2.3760 


2.6730 


0.4384 


15 


0.2972 


o.5945 


0.8917 


1. 1889 


1. 4861 


1.7834 


2.0806 


2.3778 


2.6751 


0.4388 


16 


0.2975 


o.5949 


0.8924 


1. 1898 


T.4873 


1.7848 


2.0822 


2-3797 


2.6771 


0.4392 


17 


0.2977 


0-5954 


0.8931 


1. 1908 


1.4885 


1.7862 


2.0838 


2.3815 


2.6792 


0.4396 


18 


0.2979 


o.5959 


0.8938 


1.1917 


1.4896 


1.7876 


2.0855 


2.3834 


2.6813 


0.4400 


19 


0.2982 


0.5963 


0.8945 


1. 1926 


1.4908 


1.7890 


2.0871 


2.3853 


2.6834 


0.4404 


20 


0.2984 


0.5968 


0.8952 


1. 1936 


1.4920 


1.7904 


2.0887 


2.3871 


2.6855 


0.4407 


21 


0.2986 


o.5973 


0.8959 


1 -1945 


I-493 1 


1.7918 


2.0904 


2.3890 


2.6876 


0.4411 


22 


0.2989 


o.5977 


0.8966 


I-I954 


1-4943 


1-7932 


2.0920 


2.3909 


2.6897 


0.4415 


23 


0.2991 


0.5982 


0.8973 


1. 1964 


1-4955 


1.7946 


2.0936 


2.3927 


2.6918 


0.4419 


24 


o.2993 


0.5987 


0.8980 


I-I973 


1.4966 


1.7960 


2.0953 


2.3946 


2.6939 


0.4423 


25 


0.2996 


0.5991 


0.8987 


1. 1982 


1.4978 


1-7974 


2.0969 


2.3965 


2.6960 


0.4427 


26 


0.2998 


0.5996 


0.8994 


1. 1992 


1.4989 


1.7987 


2.0985 


2.3983 


2.6981 


0.4431 


27 


0.3000 


0.6000 


0.9001 


I.20OI 


1. 5001 


1. 8001 


2.IODI 


2.4002 


2.7002 


0.4435 


28 


0.3003 


0.6005 


0.9008 


1. 20I0 


1-5013 


1.8015 


2.IOI8 


2.4020 


2.7023 


0.4438 


29 


0.3005 


0.6010 


0.9015 


I.2020 


1.5024 


1.8029 


2.IO34 


2.4039 


2.7044 


0.4442 


30 


0.3007 


0.6014 


0.9022 


I.2O29 


1.5036 


1.8043 


2.IO5O 


2.4058 


2.7065 


0.4446 


3i 


0.30C9 


0.6019 


0.9028 


I.2038 


1-5047 


1.8057 


2. I066 


2.4076 


2.7085 


0.4450 


32 


0.3012 


0.6024 


0.9035 


I.2047 


1-5059 


1. 8071 


2. I083 


2.4094 


2.7106 


C4454 


33 


0.3014 


0.6028 


0.9042 


I.2056 


1. 5071 


1.8085 


2.IO99 


2.4113 


2.7127 


0.4458 


34 


0.3016 


0.6033 


0,9049 


I.2066 


1.5082 


1.8099 


2.III5 


2.4132 


2.7148 


0.4462 


35 


0.3019 


0.6038 


0.9056 


I.2075 


1.5094 


1.8113 


2. 1 132 


2.4150 


2.7169 


0.4466 


36 


0.3021 


0.6042 


0.9063 


I.2084 


1-5105 


1. 8127 


2. 1 148 


2.4169 


2.7190 


0.4469 


37 


0.3023 


0.6047 


0.9070 


I.2094 


i-5ii7 


1. 8140 


2. 1 164 


2.4187 


2.7211 


o.4473 


38 


0.3026 


0.6051 


0.9077 


I.2I03 


1.5128 


1.8154 


2.Il8o 


2.4206 


2.7231 


0.4477 


39 


0.3028 


0.6056 


0.9084 


I.2II2 


1. 5140 


1.8168 


2. 1 196 


2.4214 


2.7252 


0.4481 


40 


0.3030 


0.6061 


0.9091 


I.2I2I 


1-5152 


1. 8182 


2.I2I2 


2.4242 


2.7273 


0.4485 


4i 


0.3033 


0.6065 


0.9098 


1. 2130 


1-5163 


1. 8196 


2.1228 


2.4261 


2.7294 


0.4489 


42 


0.3035 


0.6070 


0.9105 


1. 2140 


I-5I75 


1. 8210 


2.1245 


2.4280 


2.7315 


0.4493 


43 


0-3037 


0.6074 


0.9112 


I.2149 


1.5186 


1.8223 


2.126l 


2.4298 


2-7335 


0.4496 


44 


0.3040 


0.6079 


0.9119 


I.2I58 


1.5198 


1.8237 


2.1277 


2.4316 


2.7356 


0.4500 


45 


0.3042 


0.6084 


0.9125 


I. 2167 


1.5209 


1.8251 


2.1293 


2-4334 


2.7376 


0.4504 


46 


0.3044 


0.6088 


0.9132 


1. 2176 


1. 5221 


1.8265 


2.I309 


2-4353 


2.7397 


0.4508 


47 


0.3046 


0.6093 


0.9139 


I.2I86 


1.5232 


1.8279 


2.1325 


2.4372 


2.7418 


0.4512 


48 


0.3049 


0.6098 


0.9146 


I. 2195 


1-5244 


1.8293 


2.T34I 


2.4390 


2-7439 


0.4516 


49 


0.3051 


0.6102 


0.9I53 


I.2204 


1-5255 


1.8306 


2.1357 


2.4408 


2-7459 


0.4520 


50 


0-3053 


0.6107 


0.9160 


1. 2214 


1.5267 


1.8320 


2.1374 


2.4427 


2. 7480 


0.4524 


5i 


0.3056 


0.6111 


0.9167 


1.2223 


1.5278 


1.8334 


2.I39O 


2.4446 


2.7501 


0.4527 


52 


0.3058 


0.6116 


0.9174 


1.2232 


1.5290 


1.8348 


2. I406 


2.4464 


2.7521 


o.453i 


53 


0.3060 


0.6120 


0.9181 


1. 224I 


1-5301 


1.8361 


2.1422 


2.4482 


2.7542 


o.4535 


54 


0.3063 


0.6125 


0.9188 


I.225O 


I.53I3 


1.8375 


2.I438 


2.4500 


2.7563 


o.4539 


55 


0.3065 


0.6130 


0.9194 


I.2259 


1.5324 


1.8389 


2.I454 


2.4518 


2.7583 


o.4543 


56 


0.3067 


0.6134 


0.9201 


T.2268 


I-5336 


1.8403 


2. I47O 


2-4537 


2.7604 


o.4547 


57 


0.3069 


0.6139 


0.9208 


I.2278 


1-5347 


1. 8416 


2.I486 


2-4555 


2.7625 


o.455i 


58 


0.3072 


0.6143 


0.9215 


I.2287 


1-5358 


1.8430 


2.I502 


2-4574 


2.7645 


o.4555 


59 


0.3074 


0.6148 


0.9222 


I.2296 


I-5370 


1.8444 2.1518 


2.4592 


2.7666 


o.4558 


60 



126 DISTANCES. 


19° 


oo 


1 


3 


3 


4 


5 


6 


7 


8 


9 


a 


0.8928 


1.7855 


2.6783 


3-57IO 


4.4638 


5.3565 


6.2493 


7. 1420 


8.0348 


1-3237 


OI 


0.8926 


1.7851 


2.6777 


3.5703 


4.4629 


5-3554 


6.2480 


7. 1406 


8.0332 


1.3236 


02 


0.8924 


1.7848 


2.6772 


3-5696 


4.4620 


5-3544 


6.2468 


7.1392 


8.0316 


1.3234 


03 


0.8922 


1.7844 


2.6766 


3.5689 


4. 461 1 


5-3533 


6.2455 


7-1377 


8.0299 


I-3233 


04 


0.8920 


1. 7841 


2.6761 


3-568I 


4.4602 


5-3522 


6.2443 


7-1363 


8.0283 


1.3232 


05 


0.8919 


I-7837 


2.6756 


3.5674 


4-4593 


5.35II 


6.2430 


7-1349 


8.0267 


1.3230 


06 


0.8917 


I-7834 


2.6750 


3.5667 


4.4584 


5- 350i 


6.2417 


7.1334 


8.0251 


1.3229 


07 


0.8915 


1.7830 


2.6745 


3.5660 


4-4575 


5-34SO 


6.2405 


7. 1320 


8.0235 


1.3228 


08 


0.8913 


1.7826 


2.6740 


3.5653 


4.4566 


5-3479 


6.2392 


7-1305 


8.0219 


1.3226 


09 


0.891 1 


1.7823 


2.6734 


3.5646 


4-4557 


5.3468 


6.2380 


7-1291 


8.0203 


1.3225 


10 


0.8910 


1. 7819 


2.6729 


3.5638 


4.4548 


5-3458 


6.2367 


7.1277 


8.0186 


1.3224 


II 


0.8908 


1. 7816 


2.6723 


3-5631 


4-4539 


5-3447 


6.2354 


7.1262 


8.0170 


1.3222 


12 


0.8906 


1. 7812 


2.6718 


3.5624 


4-4530 


5-3436 


6.2342 


7.1248 


8.0154 


1. 3221 


13 


0.8904 


1.780S 


2.6712 


3-5617 


4.4521 


5-3425 


6.2329 


7.1233 


8.0137 


1.3220 


14 


0.8902 


1.7805 


2.6707 


3-5609 


4.4512 


5-34I4 


6.2316 


7-1219 


8.0121 


1.3218 


15 


0.8901 


1. 7801 


2.6702 


3.5602 


4-4503 


5-3403 


6.2304 


7.1204 


8.0105 


1-3217 


16 


0.8899 


1.7797 


2.6696 


3-5595 


4.4494 


5-3392 


6.2291 


7. 1 190 


8.0089 


1-3215 


17 


0.8897 


1-7794 


2.6691 


3.5588 


4.4485 


5.3382 


6.2279 


7."75 


8.0072 


1-3214 


18 


0.8895 


1.7790 


2.6685 


3.558o 


4.4476 


5-3371 


6.2266 


7.1161 


8.0056 


1.3213 


19 


0.8893 


1.7787 


2.6680 


3-5573 


4.4467 


5-336o 


6.2253 ! 7-1*46 


8.0040 


1.3211 


20 


0.8892 


I-7783 


2.6675 


3-5566 


4.4458 


5-3349 


6.2241 7. 1 132 


8.0024 


1. 3210 


21 


0.8890 


1-7779 


2.6669 


3-5559 


4.4448 


5-333^ 


6.2228 


7..1117 


8.C007 


1.3209 


22 


0.8888 


1.7776 


2.6664 


3-5551 


4-4439 


5-3327 


6.2215 


7. 1 103 


7.9991 


1.3207 


23 


0.8886 


1.7772 


2.6658 


3-5544 


I 4.4430 


5-33i6 


6.2202 


7.10S8 


7-9974 


1.3206 


24 


0.8884 


1.7768 


2.6653 


3-5537 


! 4-4421 


5.3305 


6.2190 


7.1074 


7-9958 


1-3205 


25 


0.8882 


1.7765 


2.6647 


3-5530 


4.4412 


5.3294 


6.2177 


7-1059 


7-9942 


1.3203 


26 


0.8881 


1. 7761 


2.6642 


3-5522 


4.4403 


5.3283 


6.2164 


7-1045 


7-9925 


1.3202 


27 


0.8879 


1.7758 


2.6636 


3-55*5 


4-4394 


5.3273 


6.2151 


7.1030 


7.9909 


1. 3201 


28 


0.8877 


1-7754 


2.6631 


3-55o8 


4-4385 


5.3262 


6.2139 


7.1015 


7-9S92 


1-3199 


29 


0.8875 


I-7750 


2.6625 


3-5500 


4-4376 


5-3251 


6.2126 


7.1001 


7.9876 


1.3198 


30 


0.8873 


1-7747 


2.6620 


3-5493 


4.4366 


5-3240 


6.2113 


7.09S6 


7.9860 


I.3I97 


31 


0.8871 


1-7743 


2.6614 


3.5486 


: 4-4357 


5-3229 


6.2100 


7.0972 


7.9843 


I.3I95 


32 


0.8870 


1-7739 


2.6609 


3o478 


4.4348 


5.3218 


6.2087 


7.0957 7.9S27 


I-3I94 


33 


0.8868 


I.7736 


2.6603 


3.5471 


4-4339 


5-3207 


6.2075 


7.0942 7.9810 


I.3I93 


34 


0.8866 


1-7732 


2.6598 


3.5464 


1 4-4330 


5-3196 


6.2062 


7.092S 


7-9794 


1.3191 


35 


0.8864 


1.7728 


2.6592 


3-5456 


i 4-4321 


5-3i85 


6.2049 


7-0913 


7.9777 


1.3190 


36 


0.8862 


I.7725 


2.65S7 


3-5449 


4-43" 


5-3174 


6. 2036 


7.0S9S 


7.9761 


1.3189 


37 


0.8860 


1. 7721 


2.6581 


3-5442 


4.4302 


5-3163 


6.2023 


7.0884 


7-9744 


1.31S7 


38 


0.8859 


1.7717 


2.6576 


3-5434 


4.4293 


5-3I52 


6.2010 


7.0869 


7.9727 


1.3186 


39 


0.8857 


1.7714 


2.6570 


3.5427 


4.4284 


5-3i4i 


6.1997 


7-0854 


7.9711 


1.3185 


40 


0.8855 


1. 7710 


2.6565 


3-5420 


4.4275 


5-3I30 


6.19S5 


7.0S40 


7.9694 


1.31S3 


4i 


0.8853 


1.7706 


2.6559 


3-5412 


4.4265 


5-3ii9 


6.1972 


7.0825 


7.9678 


1.3182 


42 


0.8851 


1.7702 


2.6554 


3.5405 


4.4256 


5-3 J 07 


6.1959 7.0S10 


7.9661 


1.31S1 


43 


0.8849 


1.7699 


2.6548 


3.5398 


4.4247 


5-3006 


6.1946 7.0795 


7-9645 


i.3i79 


44 


0.8848 


1-7695 


2.6543 


3-5390 


4.4238 


5-3085 


6.1933 


7.0780 


7.9628 


1.3178 


45 


0.8846 


1. 7691 


2.6537 


3.5383 


4.4229 


5-3074 


6. 1920 


7.0766 


7.961 1 


I.3I77 


46 


0.8844 


1.7688 


2.6532 


3-5375 


4.4219 


5-3063 


6.1907 


7.0751 


7-9595 


I.3I75 


47 


0.8842 


1.7684 


2.6526 


3-536S 


j 4.4210 


5-3052 


6.1S94 


7-0736 


7.9578 


i.3*74 


48 


0.8840 


1.7680 


2.6520 


3.536i 


j 4.4201 


5-3041 


6.1SS1 


7.0721 


7.956i 


*.3!73 


49 


0.8838 


1.7677 


2.6515 


3-5353 


i 4.4192 


5-3030 


6.1S6S 


7.0706 


7-9545 


1.3171 


50 


0.8836 


1-7673 


2.6509 


3.5346 


4.41S2 


5-3019 


6.1S55 


7.0692 


7-9528 


1-3170 


5i 


0.8835 


1.7669 


2.6504 


3.533S 


4-4173 


5.30CS 


6.1S42 


7.0677 


7-95H 


1. 3169 


52 


0.8833 


1.7665 


2.649S 


3-5331 


4.4164 


5.2996 


6.1S29 


7.0662 


7-9495 


1.3167 


53 


0.SS31 


1.7662 


2.6493 


3.5324 


4-4I54 


5-29S5 


6.1816 


7.0647 


7-947S 


1.3166 


54 


0.8S29 


1.7658 


2.64S7 


3.53i6 


4.4145 


5-2974 


6.1S03 


7.0632 


7.9461 


1.3165 


55 


0.8827 


1.7654 


2.64S1 


3.5309 


4.4136 


5-2963 


6.1790 


7.0617 


7-9444 


1.3163 


56 


0.8825 


1. 7651 


2.6476 


3- 530i 


4.4127 


5-2952 


6.1777 


7.0602 


7.9428 


1.3162 


57 


0.SS23 


1.7647 


2.6470 


3-5294 


4.4117 


5.2941 


6.1764 


7.C5SS 


7.9411 


1.3160 


5S 


0.8822 


1-7643 


2.6465 


3.52S6 


4.410S 


5.2929 


6. 1 75 1 


7-0573 


7-9394 


1. 3i59 


59 


0.SS20 


1.7639 


2.6459 


3.5279 


4.4099 


5.2918 


6.173S 


7-055S 


7-9377 


1. 3158 


60 


0.SS18 


1.7636 


2.6454 


3-5271 


4.40S9 


5-2907 


6.1725 


7-0543 


7-936i 


i- 3*56 



19° 




HEIGHTS. 127 


1 


2 


3 


4 


5 


6 


7 


8 


9 


b 


00 


0.3074 


0.6148 


0.9222 


1.2296 


1-5370 


1.8444 


2.1518 


2.4592 


2.7666 


o.4558 


0.3076 


0.6153 


0.9229 


1.2305 


1. 538i 


1.8458 


2.1534 


2.4610 


2.7687 


0.4562 


01 


0.3079 


0.6157 


0.9236 


1.2314 


1-5393 


1.8472 


2.1550 


2.4629 


2.7707 


0.4566 


02 


0.308 1 


0.6162 


0.9243 


1.2324 


1.5404 


1.8485 


2.1566 


2.4647 


2.7728 


0.4570 


03 


0.3083 


0.6166 


0.9249 


1-2333 


1. 5416 


1.8499 


2.1582 


2.4665 


2.7748 


o.4573 


04 


0.3085 


0.6171 


0.9256 


1.2342 


L5427 


1.8512 


2.1598 


2.4683 


2.7769 


o.4577 


°l 


0.3088 


0.6175 


0.9263 


1-2351 


1-5439 


1.8526 


2.1614 


2.4702 


2.7789 


0.4581 


06 


0.3090 


0.6180 


0.9270 


1.2360 


I-5450 


1.8540 


2. 1630 


2.4720 


2.7810 


0.4585 


07 


0.3092 


0.6185 


0.9277 


1.2369 


1. 5461 


1-8554 


2.1646 


2.4738 


2.7831 


0.4589 


08 


0.3095 


0.6189 


0.9284 


1.2378 


1-5473 


1.8568 


2.1662 


2-4757 


2.7851 


0-4593 


09 


0.3097 


0.6194 


0.9290 


1.2387 


1.5484 


1.8581 


2.1678 


2-4775 


2.7871 


0.4596 


10 


0.3099 


0.6198 


0.9297 


1.2396 


1-5495 


1.8595 


2.1694 


2-4793 


2.7892 


0.4600 


n 


0.3101 


0.6203 


0.9304 


1.2406 


I-5507 


1.8608 


2.1710 


2.481 1 


2.79*3 


0.4604 


12 


0.3104 


0.6207 


0.931 1 


1.2415 


i.55i8 


1.8622 


2.1726 


2.4830 


2-7933 


0.4608 


J3 


0.3106 


0.6212 


0.9318 


1.2424 


1 -5530 


1.8636 


2.1742 


2.4848 


2-7953 


0.4612 


14 


0.3108 


0.6216 


0.9325 


L2433 


*-554i 


1.8649 


2.1758 


2.4866 


2.7974 


0.4616 


15 


0.3110 


0.6221 


o.933i 


1.2442 


1-5552 


1.8663 


2.1773 


2.4884 


2-7994 


0.4619 


16 


0.3113 


0.6226 


o.9338 


1.2451 


1.5564 


1.8677 


2.1789 


2.4902 


2.8015 


0.4623 


17 


0.3115 


0.6230 


o.9345 


1.2460 


1-5575 


1.8690 


2.1805 


2.4920 


2.8035 


0.4627 


18 


0.31 17 


0.6235 


o.9352 


1.2469 


1.5587 


1.8704 


2.1821 


2.4938 


2.8056 


0.4631 


19 


0.3120 


0.6239 


o.9359 


1.2478 


1.5598 


1.8718 


2.1837 


2.4957 


2.8076 


0.4635 


20 


0.3122 


0.6244 


0.9365 


1.2487 


1.5609 


I-873I 


2.1853 


2-4975 


2.8096 


0.4639 


21 


0.3124 


0.6248 


0.9372 


1.2496 


1.5620 


1.8745 


2.1869 


2-4993 


2.8117 


0.4642 


22 


0.3126 


0.6253 


o.9379 


1.2505 


1.5632 


1.8758 


2.1885 


2.501 1 


2.8137 


0.4646 


23 


0.3129 


0.6257 


0.9386 


1. 2514 


1-5643 


1.8772 


2.1900 


2.5029 


2.8157 


0.4650 


24 


0.3131 


0.6262 


o.9393 


1.2524 


1-5654 


1.8785 


2.1916 


2.5047 


2.8178 


0.4654 


25 


o.3i33 


0.6266 


0.9399 


L2533 


1.5666 


1.8799 


2.1932 


2.5065 


2.8198 


0.4658 


26 


0.3135 


0.6271 


0.9406 


1.2542 


I-5677 


1. 8812 


2. 1948 


2.5083 


2.8219 


0.4662 


27 


0.3138 


0.6275 


0.9413 


i.2S5i 


1.5688 


1.8826 


2. 1964 


2.510T 


2.8239 


0.4665 


i 28 


0.3140 


0.6280 


0.9420 


1.2560 


1.5700 


1.8839 


2.1979 


2.5119 


2.8259 


0.4669 


'29 


0.3142 


0.6284 


0.9427 


1.2569 


I-57 11 


1.8853 


2.1995 


2.5138 


2.8280 


0.4673 


30 


0.3144 


0.6289 


o-9433 


1.2578 


1.5722 


1.8867 


2.2011 


2.5156 


2.83CO 


0.4677 


k 


0.3147 


0.6293 


0.9440 


1.2587 


1-5734 


1.8880 


2.2027 


2.5174 


2.8320 


0.4681 


32 


0.3149 


0.6298 


0.9447 


1.2596 


1-5745 


1.8894 


2.2043 


2.5192 


2.8341 


0.4685 


33 


0.3151 


0.6302 


0.9454 


1.2605 


1.5756 


1.8907 


2.2058 


2.5210 


2.8361 


0.4689 


34 


o.3i53 


0.6307 


0.9460 


1. 2614 


1-5767 


1. 8921 


2.2074 


2.5228 


2.8381 


0.4692 


35 


0.3156 


O.631 1 


0.9467 


1.2623 


1-5779 


1.8934 


2.2090 


2.5246 


2.8401 


0.4696 


36 


0.3158 


0.6316 


0.9474 


1.2632 


I.5790 


1.8948 


2.2106 


2.5264 


2.8422 


0.4700 


37 


0.3160 


0.6320 


0.9481 


1. 2641 


T.5801 


1. 8961 


2.2122 


2.5282 


2.8442 


0.4704 


38 


0.3162 


0.6325 


0.9487 


1.2650 


1.5812 


1.8975 


2.2137 


2.5300 


2.8462 


0.4708 


39 


0.3165 


0.6329 


0.9494 


1.2659 


1.5824 


1.8988 


2.2153 


2.5318 


2.8482 


0.4712 


40 


0.3167 


C6334 


0.9501 


1.2668 


1.5835 


1.9002 


2.2169 


2.5336 


2.8503 


o.47i5 


41 


0.3169 


0.6338 


0.9508 


1.2677 


1.5846 


1.9015 


2.2184 


2-5354 


2.8523 


0.4719 


42 


0.3171 


0.6343 


0.9514 


1.2686 


1.5857 


1.9029 


2.2200 


2.5372 


2.8543 


0.4723 


43 


0.3174 


O.6347 


0.9521 


1.2695 


1.5868 


1.9042 


2.2216 


2.5390 


2.8563 


0.4727 


44 


0.3176 


0.6352 


0.9528 


1.2704 


1.5880 


I-9055 


2.2231 


2.5407 


2.8583 


o.473i 


45 


0.3178 


0.6356 


o.9535 


1.2713 


1.5891 


1.9069 


2.2247 


2.5425 


2.8604 


C4735 


46 


0.3180 


0.6361 


0.9541 


1.2722 


1.5902 


1.9082 


2.2263 


2-5443 


2.8624 


o.4739 


47 


0.3183 


0.6365 


0.9548 


1.2731 


I-59I3 


1.9096 


2.2279 


2.5461 


2.8644 


0.4742 


48 


0.3185 


0.6370 


o.9555 


1.2740 


1.5924 


1.9109 


2.2294 


2-5479 


2.8664 


0.4746 


49 


0.3187 


0.6374 


0.9561 


1.2748 


1-5936 


1.9123 


2.2310 


2-5497 


2.S684 


c.4750 


50 


0.3189 


0.6379 


0.9568 


1.2757 


1-5947 


1.9136 


2.2326 


2.5515 


2.8704 


o.4754 


51 


0.3192 


0.6383 


o.9575 


1.2766 


1.5958 


1. 9150 


2.2341 


2-5533 


2.8724 


0.4758 


52 


0.3194 


0.6388 


0.9581 


1.2775 


1.5969 


1. 9163 


2.2357 


2.5551 


2.8744 


0.4761 


53 


0.3196 


0.6392 


0.9588 


1.2784 


1.5980 


1.9177 


2.2373 


2.5569 


2.8765 


0.4765 


54 


0.3198 


0.6397 


o.9595 


1.2793 


1. 5991 


1. 9190 


2.2388 


2.5586 


2.8785 


0.4769 


55 


0.3201 


0.6401 


0.9602 


1.2802 


1.6003 


1.9203 


2.2404 


2.5604 


2.8805 


o.4773 


56 


0.3203 


0.6405 


0.9608 


1.2811 


1. 6014 


1. 9216 


2.2419 


2.5622 


2.8825 


o.4777 


57 


0.3205 


0.6410 


0.9615 


1.2820 


1.6025 


1-9230 


2.2435 


2.5640 


2.8845 


0.47S1 


58 


0.3207 


0.6414 


0.9622 


1.2829 


1.6036 


1.9243 


2.2450 


2.5658 


2.8865 


0.4784 


59 


c.3209 


0.6419 


0.9628 


1.2838 


1.6047 


1.5257 


2.2465 


2.5675 


2.8885 


0.4788 


60 



